Find the derivative of the function f(x) = x 2 – 2x.
-
Upload
sibyl-eaton -
Category
Documents
-
view
243 -
download
0
Transcript of Find the derivative of the function f(x) = x 2 – 2x.
![Page 1: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/1.jpg)
Find the derivative of the function f(x) = x2 – 2x
0
( ) ( )lim'( )h
f x h f x
hf x
2 2
0
( ) 2( ) ( 2 )limh
x h x h x x
h
2 2 2
0
2 2 2 2limh
x xh h x h x x
h
2
0
2 2limh
xh h h
h
0lim 2 2h
x h
2 2x
![Page 2: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/2.jpg)
State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions.
Objective – To be able to find the derivative of a function.
![Page 3: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/3.jpg)
RULE 1 Derivative of a Constant Function
If f has the constant value f(x) = c, then
( ) 0df d
cdx dx
If the derivative of a function is its slope, then for a constant function, the derivative must be zero.
![Page 4: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/4.jpg)
Example 1a
Find the derivative of f(x) = 8
df
dx(8)
d
dx 0
Example 1b
Find the derivative of
df
dx 2
d
dx
0
( )2
f x
![Page 5: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/5.jpg)
RULE 2 Power Rule for Positive Integers
If n is a positive integer, then
1n ndx nx
dx
![Page 6: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/6.jpg)
In the Warm-Up we saw that if , .
2 2y x x This is part of a pattern.
1n ndx nx
dx
examples:
4f x x
34f x x
8y x78y x
power rule
2 2y x
![Page 7: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/7.jpg)
y x
1
21
2y x
1
2y x
1
2
1
2
y
x
1
2y
x
![Page 8: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/8.jpg)
RULE 3 Constant Mulitple Rule
If u is a differentiable function of x, and c is a constant, then
d ducu c
dx dx
![Page 9: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/9.jpg)
d ducu c
dx dx
examples:
1n ndcx cnx
dx
constant multiple rule:
5 4 47 7 5 35d
x x xdx
![Page 10: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/10.jpg)
RULE 4 The Sum Rule
If f and g are both differentiable, then
( )d du dv
u vdx dx dx
![Page 11: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/11.jpg)
Example
Find the derivative of y = x4 + 12x
4( ) (12 )dy d d
x xdx dx dx
34 12x
![Page 12: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/12.jpg)
RULE 5 The Difference Rule
If f and g are both differntiable, then
( )d du dv
u vdx dx dx
![Page 13: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/13.jpg)
Example
Find the derivative of y = x3 – 3x
3( ) (3 )dy d d
x xdx dx dx
23 3x
![Page 14: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/14.jpg)
RULE 6 The Derivative of the Natural Exponential Function
( )x xde e
dx
![Page 15: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/15.jpg)
Pg. 191
3 – 31 odd
![Page 16: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/16.jpg)
Find the derivative of the function f(x) = 3x2 – 5x + 1
6 5x
![Page 17: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/17.jpg)
State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions.
Objective – To be able to find the derivative of a function.
![Page 18: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/18.jpg)
Derivative of Sine and Cosine Functions:
sin cosd
x xdx
cos sind
x xdx
![Page 19: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/19.jpg)
THE PRODUCT RULE:
( ) ( ) ( ) ( ) ( ) ( )d
f x g x f x g x g x f xdx
![Page 20: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/20.jpg)
Example 1
Find the derivative of f(x) = (2x+5)(3+4x)
( ) ( ) ( ) ( ) ( )f x f x g x g x f x
( ) (2 5)(4) (3 4 )(2)f x x x
( ) 16 26f x x
f(x) g(x)
( ) 8 20 6 8f x x x
![Page 21: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/21.jpg)
Example 2
Find the derivative of f(x) = (4x3)(sin x)
( ) ( ) ( ) ( ) ( )f x f x g x g x f x
3 2( ) (4 )(cos ) (sin )(12 )f x x x x x
3 2( ) 4 cos 12 sinf x x x x x
f(x) g(x)
![Page 22: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/22.jpg)
Example 3
Find the derivative of y = (5x2)(cos x) + (3x)(sinx)
( ) ( ) ( ) ( )y f x g x g x f x
2(5 )( sin ) (cos )(10 ) (3 )(cos ) (sin )(3)y x x x x x x x
25 sin 13 cos 3siny x x x x x
25 sin 10 cos 3 cos 3siny x x x x x x x
![Page 23: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/23.jpg)
3-25 odd
![Page 24: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/24.jpg)
Find the derivative of the function f(x) = (x – 4)(x + 3)
( ) ( ) ( ) ( ) ( )f x f x g x g x f x
( ) ( 4)(1) ( 3)(1)f x x x
( ) 2 1f x x
( ) 4 3f x x x
![Page 25: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/25.jpg)
State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions.
Objective – To be able to find the derivative of a function.
![Page 26: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/26.jpg)
THE QUOTIENT RULE:
2
( ) ( ) ( ) ( ) ( )
( ) ( )
d f x g x f x f x g x
dx g x g x
![Page 27: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/27.jpg)
Example 1
Find the derivative of
2
( ) ( ) ( ) ( )( )
( )
g x f x f x g xf x
g x
2
( )(2) (2 1)(1)( )
x xf x
x
2
1( )f x
x
f(x)
g(x)
2
2 2 1( )
x xf x
x
2 1( )
xf x
x
![Page 28: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/28.jpg)
Example 2
Find the derivative of
2
( ) ( ) ( ) ( )( )
( )
g x f x f x g xf x
g x
2
(5 2)(3) (3 4)(5)( )
(5 2)
x xf x
x
2
26( )
(5 2)f x
x
f(x)
g(x)
2
15 6 15 20( )
(5 2)
x xf x
x
3 4( )
5 2
xf x
x
![Page 29: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/29.jpg)
Example 3
Find the derivative of
2
( ) ( ) ( ) ( )( )
( )
g x f x f x g xf x
g x
3 2
3 2
( )( sin ) (cos )(3 )( )
( )
x x x xf x
x
4
sin 3cos( )
x x xf x
x
f(x)
g(x)
2
6
( sin 3cos )( )
x x x xf x
x
3
cos( )
xf x
x
![Page 30: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/30.jpg)
3 – 25 odd
![Page 31: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/31.jpg)
State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions.
Objective – To be able to find the derivative of a function.
![Page 32: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/32.jpg)
Derivatives of the remaining trig functions:
sin cosd
x xdx
cos sind
x xdx
2tan secd
x xdx
2cot cscd
x xdx
sec sec tand
x x xdx
csc csc cotd
x x xdx
![Page 33: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/33.jpg)
Example 1
Find the derivative of y = (sec x)(tan x)
( ) ( ) ( ) ( )y f x g x g x f x
2(sec )(sec ) (tan )(sec tan )y x x x x x
32sec secy x x
3 2sec sec tany x x x 3 2sec sec (sec 1)y x x x
3 3sec sec secy x x x
![Page 34: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/34.jpg)
Higher Order Derivatives:
dyy
dx is the first derivative of y with respect to x.
2
2
dy d dy d yy
dx dx dx dx
is the second derivative.
(y double prime)
dyy
dx
is the third derivative.
4 dy y
dx is the fourth derivative.
We will learn later what these higher order derivatives are used for.
![Page 35: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/35.jpg)
4 210 33 15y x x
' 340 66y x x '' 2120 66y x
''' 240y x
4 240y
![Page 36: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/36.jpg)
WS
![Page 37: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/37.jpg)
1) sin cosd
x xdx
2) cos sind
x xdx
23) tan secd
x xdx
24) cot cscd
x xdx
5) sec sec tand
x x xdx
6) csc csc cotd
x x xdx
Find the Derivative:
![Page 38: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/38.jpg)
State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of algebraic, trig, exponential, and logarithmic functions.
Objective – To be able to find the derivative of a function.
![Page 39: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/39.jpg)
So far we have been memorizing the derivatives of the trig functions. And today we will be investigating this further. This will help us as we go into the next section of using the Chain Rule.
![Page 40: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/40.jpg)
2) Find the derivative of f(x) = (x)(sin x)
( ) ( ) ( ) ( ) ( )f x f x g x g x f x
( ) ( )(cos ) (sin )(1)f x x x x
( ) cos sinf x x x x
![Page 41: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/41.jpg)
ON WHITE BOARD
a) Find the derivative of f(x) = (x)(cos x)
b) Find the derivative of f(x) = (x)(tan x)
( ) ( ) ( ) ( ) ( )f x f x g x g x f x ( ) ( )( sin ) (cos )(1)f x x x x
( ) sin cosf x x x x
( ) ( ) ( ) ( ) ( )f x f x g x g x f x 2( ) ( )(sec ) (tan )(1)f x x x x
2( ) sec tanf x x x x
![Page 42: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/42.jpg)
4) Find the derivative of y = 2 csc x + 5 cos x
2( csc cot ) 5( sin )y x x x
2csc cot 5siny x x x
![Page 43: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/43.jpg)
ON WHITE BOARD
a) Find the derivative of y = 4 sec x + 3 sin x
b) Find the derivative of y = 7 cot x + 2 tan x
4(sec tan ) 3(cos )y x x x
4sec tan 3cosy x x x
2 27( csc ) 2(sec )y x x
2 27csc 2secy x x
![Page 44: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/44.jpg)
10) Find the derivative of
2
( ) ( ) ( ) ( )( )
( )
g x f x f x g xf x
g x
2
( cos )(cos ) (1 sin )(1 sin )
( cos )
x x x x xy
x x
2
cos
( cos )
x xy
x x
2 2
2
cos cos (1 sin )
( cos )
x x x xy
x x
1 sin
cos
xy
x x
2 2
2
cos cos 1 sin
( cos )
x x x xy
x x
1
![Page 45: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/45.jpg)
ON WHITE BOARD
12) 2
( ) ( ) ( ) ( )( )
( )
g x f x f x g xf x
g x
2
2
(sec )(sec ) (tan 1)(sec tan )
(sec )
x x x x xy
x
2 2sec tan tan
sec
x x xy
x
3 2
2
sec (sec tan sec tan )
sec
x x x x xy
x
tan 1
sec
xy
x
3 2
2
sec sec tan sec tan
sec
x x x x xy
x
1 tan
sec
xy
x
2 21 tan secx x
![Page 46: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/46.jpg)
Pg. 216
1 – 15 odd
![Page 47: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/47.jpg)
cscy xcsc coty x x
2(csc )( csc ) (cot )( csc cot )y x x x x x
2csc(2csc 1)y x
32csc cscy x x
Find y´´
3 2csc csc cot )y x x x 3 2csc csc coty x x x
2 2csc (csc cot )y x x x 2 2csc (csc csc 1)y x x x
![Page 48: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/48.jpg)
![Page 49: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/49.jpg)
State Standard – 5.0 Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
Objective – To be able to use the Chain Rule to solve applications.
![Page 50: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/50.jpg)
We now have a pretty good list of “shortcuts” to find derivatives of simple functions.
Of course, many of the functions that we will encounter are not so simple. What is needed is a way to combine derivative rules to evaluate more complicated functions.
![Page 51: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/51.jpg)
Consider a simple composite function:
6 10y x
2 3 5y x
If 3 5u x
then 2y u
6 10y x 2y u 3 5u x
6dy
dx 2
dy
du 3
du
dx
dy dy du
dx du dx
6 2 3
![Page 52: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/52.jpg)
one more:29 6 1y x x
23 1y x
If 3 1u x
3 1u x
18 6dy
xdx
2dy
udu
3du
dx
dy dy du
dx du dx
2y u
2then y u
29 6 1y x x
2 3 1dy
xdu
6 2dy
xdu
18 6 6 2 3x x This pattern is called the chain rule.
![Page 53: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/53.jpg)
The Chain Rule can be written either in Leibniz notation:
Or in Prime Notation:
( ) ( ( )) ( )F x f g x g x
dy dy du
dx du dx
![Page 54: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/54.jpg)
Example 1
Find of
4 3u x 9y u
dy
dx
827(4 3 )dy
xdx
9(4 3 )y x
dy du
du dx
89dy
udu
3du
dx
89u ( 3) 827u
![Page 55: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/55.jpg)
On White Board
Find of
3 45u x x 7y u
dy
dx
3 4 6 2 37(5 ) (15 4 )dy
x x x xdx
3 4 7(5 )y x x
dy du
du dx
67dy
udu
2 315 4du
x xdx
67u 2 3(15 4 )x x
![Page 56: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/56.jpg)
On White Board
Find of
43 7 5u x x 3y u
dy
dx
4 2 33(3 7 5) (12 7)dy
x x xdx
4 3(3 7 5)y x x
dy du
du dx
23dy
udu
312 7du
xdx
23u 3(12 7)x
![Page 57: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/57.jpg)
9 – 42 mult of 3
![Page 58: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/58.jpg)
secy xsec tany x x
2sec (sec ) tan (sec tan )y x x x x x
32sec secy x x
Find y´´
2 2sec (sec tan )y x x x 2 2sec (sec sec 1)y x x x
2sec (2sec 1)y x x
![Page 59: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/59.jpg)
![Page 60: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/60.jpg)
State Standard – 5.0 Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
Objective – To be able to use the Chain Rule to solve applications.
![Page 61: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/61.jpg)
The Chain Rule can be written either in Leibniz notation:
Or in Prime Notation:
( ) ( ( )) ( )F x f g x g x
dy dy du
dx du dx
![Page 62: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/62.jpg)
Example 1
Find of
sin 3u x 9y u
dy
dx
89cos (sin 3)dy
x xdx
9(sin 3)y x
dy du
du dx
89dy
udu
cosdu
xdx
89u (cos )x89cos ( )x u
![Page 63: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/63.jpg)
On White Board
Find of
3
2
xu
sin cosy u u
dy
dx
3 3 3cos sin
2 2 2
dy x x
dx
3 3sin cos
2 2
x xy
dy du
du dx
cos sindy
u udu
3
2
du
dx
(cos sin )u u 3
2
![Page 64: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/64.jpg)
On White Board
Find of
2 1u x cosy u
dy
dx
22 sin( 1)dy
x xdx
2cos( 1)y x
dy du
du dx
sindy
udu
2du
xdx
sin u (2 )x
![Page 65: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/65.jpg)
9 – 42 mult of 3
![Page 66: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/66.jpg)
Find the Derivative21) sin( 2 )x x
22) tan 4x3 23) sin(3 )x
24) cos10x25) cos x
26) cos )x37) sin(3 )x
38) cos( 4 )x x 29) sin 5x
3 210) cos( )x211) cos 4x
212) tan x213) sin )x
514) cos( )x
15) 3(tan 4 )x
16) sin 5x317) cos(2 )x
![Page 67: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/67.jpg)
secy xsec tany x x
2sec (sec ) tan (sec tan )y x x x x x
32sec secy x x
Find y´´
2 2sec (sec tan )y x x x 2 2sec (sec sec 1)y x x x
2sec (2sec 1)y x x
![Page 68: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/68.jpg)
![Page 69: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/69.jpg)
State Standard – 5.0 Students know the Chain Rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
Objective – To be able to use the Chain Rule to solve applications.
![Page 70: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/70.jpg)
Repeated Use of the Chain Rule
We sometimes have to use the Chain Rule two or more times to find a derivative.
![Page 71: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/71.jpg)
Example 1
Find the derivative of g(t) = tan (5 – sin 2t)
( ) (tan(5 sin 2 ))d
g t tdt
5 sin 2u t
dy du
du dt
2secdy
udu
(5 sin 2 )du d
tdt dt
2sec u (5 sin 2 )d
tdt
tany u
2sec (5 sin 2 )t (5 sin 2 )d
tdt
2( ) sec (5 sin 2 ) ( 2cos 2 )g t t t 2u tcos
dyu
du
2du
dt
5 siny u
![Page 72: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/72.jpg)
Example 2
Find the derivative of f(x) = cos ( sin 3x)
( ) (cos(sin 3 ))d
f x tdx
sin 3u x
dy du
du dx
sindy
udu
(sin 3 )du d
xdx dx
sin u (sin 3 )d
xdx
cosy u
sin(sin 3 )x (sin 3 )d
xdx
( ) sin(sin 3 ) (3cos3 )f x x x 3u xcos
dyu
du
3du
dx
siny u
![Page 73: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/73.jpg)
Find the Derivative
1) sin(cos(2 5))y x
3
2) 1 tan12
xy
23) 1 cos( )y x
![Page 74: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/74.jpg)
9 – 42 mult of 3
![Page 75: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/75.jpg)
Find the Derivative
2
2 2
3
2
1) tan(sin(3 5))
2) sin(cos( ))
3) cos (3 2 )
4) 3 sec3
5) 4 cos 3
y x
y x
y x x
xy
y x x
2
2 3 2 2
4
3 2
6) 1 tan( )
7) 1 cos10
8) sin ( 4)cos ( )
9) tan (3 1)
10) cos (cos (cos ))
y x
xy
y x x
y x
y x
![Page 76: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/76.jpg)
Find y´´
1) sin cosd
x xdx
2) cos sind
x xdx
23) tan secd
x xdx
24) cot cscd
x xdx
5) sec sec tand
x x xdx
6) csc csc cotd
x x xdx
Find the Derivative:
![Page 77: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/77.jpg)
State Standard – 6.0 Students use implicit differentiation in a wide variety of problems.
Objective – To be able to use implicit differentiation
![Page 78: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/78.jpg)
Implicit Differentiation
(Takes Four Steps)
1) Differentiate both sides of the equation with respect to ‘x’, treating ‘y’ as a differentiable function of ‘x’.
2) Collect the terms with the dy/dx on one side of the equation.
3) Factor out the dy/dx.
4) Solve for dy/dx.
![Page 79: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/79.jpg)
2 2 1x y This is not a function, but it would still be nice to be able to find the slope.
2 2 1d d d
x ydx dx dx
Do the same thing to both sides.
2 2 0dy
x ydx
Note use of chain rule.
2 2dy
y xdx
2
2
dy x
dx y
dy x
dx y
![Page 80: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/80.jpg)
22 siny x y
22 sind d d
y x ydx dx dx
This can’t be solved for y.
2 2 cosdy dy
x ydx dx
2 cos 2dy dy
y xdx dx
22 cosdy
xydx
2
2 cos
dy x
dx y
This technique is called implicit differentiation.
1 Differentiate both sides w.r.t. x.
2 Solve for .dy
dx
![Page 81: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/81.jpg)
Example 1
Find of
4 6 ( ) 0x y y 4x
dy
dx
2
3
xy
y
2 22 3 4x y
6y
6 ( ) 4y y x
4x
6y
![Page 82: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/82.jpg)
Example 2
Find of
2 2 ( ) 5 4( )x y y y 2x
dy
dx
5 2
2 4
xy
y
2 2 5 4x y x y
2 4y
2 ( ) 4( ) 5 2y y y x
2x
2 4y
4( )y 4( )y
(2 4) 5 2y y x
![Page 83: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/83.jpg)
Example 3
Find of
2 23 3 ( ) 3 (1)( ) (1)( )x y y x y y
2x
dy
dx
2
2
x yy
y x
3 3 3x y xy
2y x
2 2 ( ) ( )x y y x y y 2x
2y x
( )x y ( )x y2 2( ) ( )y y x y x y
3 3
2 2( )y y x x y
![Page 84: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/84.jpg)
Example 4
Find of
2 (sin )( sin )( ) (cos )(cos ) 0x y y y x
(cos )(cos )y x
dy
dx
cos cos
sin sin
y xy
x y
2sin cos 1x y
(sin )( sin )x y
(sin )( sin )( ) (cos )(cos ) 0x y y y x (cos )(cos )y x
(sin )( sin )x y
(sin )( sin )( ) (cos )(cos )x y y y x
2 2
![Page 85: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/85.jpg)
Pg. 233
1a, 2a, 3a, 5 – 9, 11, 13, 14, and 20
![Page 86: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/86.jpg)
From www.Dictionary.com
Explicit-
6.Mathematics. (of a function) having the dependent variable expressed directly in terms of the independent variables, as y = 3x + 4.
Implicit-4.Mathematics. (of a function) having the dependent variable not explicitly expressed in terms of the independent variables, as x2 + y2 = 1.
![Page 87: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/87.jpg)
Find of
6 2 ( ) 2 5( )x y y y 6x
dy
dx
2 6
2 5
xy
y
2 23 2 5x y x y
2 5y
2 ( ) 5( ) 2 6y y y x
6x
2 5y
5( )y 5( )y
(2 5) 2 6y y x
![Page 88: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/88.jpg)
State Standard – 4.4 Students derive derivative formulas and use them to find the derivatives of inverse trig functions.
Objective – To be able to take derivatives of Inverse Trig Functions.
![Page 89: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/89.jpg)
1
2
1(sin )
1
du u
dx u
1sin arcsinx x
1
2
1(cos )
1
du u
dx u
12
1(tan )
1
du u
dx u
1
2
1(csc )
1
du u
dx u u
1
2
1(sec )
1
du u
dx u u
12
1(cot )
1
du u
dx u
![Page 90: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/90.jpg)
1
2
1(sin )
1
du u
dx u
1sin x y
sinx yd d
dx dx
1 cosdy
ydx
2 2 1x b
1
cos
dy
dx y
1
y
x
b1
2 21b x 21b x
21
1
x
adj
hyp 2
1
1
dy
dx x
![Page 91: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/91.jpg)
Example 1
Find the derivative of
2 2
1( ) 2
1 ( )f x x
x
4
2( )
1
xf x
x
1 2( ) sinf x x2u x
1
2
1(sin )
1
du u
dx u
![Page 92: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/92.jpg)
Example 2
Find the derivative of
1
221
2
1 1( )
21
f x x
x
1( )
(2 2)f x
x x
1( ) tanf x x
1
2u x
12
1(tan )
1
du u
dx u
11 2( ) tanf x x
1 1( )
1 2f x
x x
1
( )(2)(1 )
f xx x
![Page 93: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/93.jpg)
Find the Derivative.1 21) cos ( )y x
1 12) cosy
x
13) sin 2y x
14) sin 1y x
15) sec (2 1)y s
16) tan 3 1y x
17) csc2
xy
12
38) siny
x
![Page 94: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/94.jpg)
In Sec. 3.2 we learned the basics of Higher Order Derivatives. So Find the first four derivatives of
23 6y x x
(4) 0y
3 23 2y x x
6 6y x
6y
First Derivative:
Second Derivative:
Third Derivative:
Fourth Derivative:
![Page 95: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/95.jpg)
Velocity and Acceleration
![Page 96: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/96.jpg)
State Standard – 7.0 Students compute derivatives of higher order.
Objective – To be able to solve problems involving multiple derivative steps.
![Page 97: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/97.jpg)
Higher Order Derivatives:
dyy
dx is the first derivative of y with respect to x.
2
2
dy d dy d yy
dx dx dx dx
is the second derivative.
(y double prime)
dyy
dx
is the third derivative.
4 dy y
dx is the fourth derivative.
We will learn later what these higher order derivatives are used for.
![Page 98: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/98.jpg)
Example 1
Find the second derivative of
2 sin 4cosy x x x
2 siny x x
(2 )(cos ) (sin )(2)y x x x
2 cos 2siny x x x
(2 )( sin ) (cos )(2) 2cosy x x x x
2 sin 2cos 2cosy x x x x
![Page 99: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/99.jpg)
Example 2
Find the second derivative of
4
2 6xy
x
2
1xy
x
2
2 2
(1)( ) ( 1)(2 )
( )
x x xy
x
2 2
4
2 2x x xy
x
2
4
2x xy
x
3 2
3 2
( 1)( ) ( 2)(3 )
( )
x x xy
x
2
( ) ( ) ( ) ( )
[ ( )]
f x g x f x g xy
g x
3 3 2
6
( 3 6 )x x xy
x
3
2x
x
3 3 2
6
3 6x x xy
x
3 2
6
2 6x xy
x
![Page 100: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/100.jpg)
Acceleration is the first derivative of Velocity.
Velocity is the first derivative of position.
Acceleration is the second derivative of position.
( ) ( )ds
v t s tdt
( ) ( ) ( )a t v t s t
( )s s t is the position function
![Page 101: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/101.jpg)
Example 3
The position of a particle is given by the equation:
Where t is measured in seconds and s in meters.
a) Find the acceleration at time t. What is the acceleration after 4 seconds.
3 2( ) 6 9s f t t t t
2( ) 3 12 9ds
v t t tdt
( ) 6 12dv
a t tdt
(4) 6(4) 12a 2
(4) 12m
as
24 12
![Page 102: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/102.jpg)
Pg. 240
5 – 15 odd, 29, 31, 43 and 44
![Page 103: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/103.jpg)
The position of a particle is given by the equation:
Where t is measured in seconds and s in meters.
a) Find the acceleration at time t. What is the acceleration after 5 seconds.
3 2( ) 2 5 4s f t t t t
2( ) 6 10 4ds
v t t tdt
( ) 12 10dv
a t tdt
(5) 12(5) 10a 2(5) 50
ma
s60 10
![Page 104: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/104.jpg)
A dynamite blast blows a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of:
Where t is measured in seconds and s in feet.
a) How high does the rock go?
2160 16 sec.s t t ft after t
( ) 160 32 / secds
v t t ftdt
0 160 32t 32 160t
( ) 45 00s ft5sect
What is the velocity at the rock’s height?25 5( ) 160( ) 516( )s
( ) 80 05 0 4 0s
![Page 105: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/105.jpg)
State Standard – 4.4 Students find the derivatives of logarithmic functions.
Objective – Students will be able to solve problems involving logarithmic functions.
![Page 106: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/106.jpg)
Derivative of Log Functions
1ln
dx
dx x
) ln8a y x1
88
yx
1ln
du u
dx u
Example 1
1y
x
2) ln( 2)b y x
2
12
2y x
x
2
2
2
xy
x
![Page 107: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/107.jpg)
Derivative of Log Functions
1ln
dx
dx x
) lna y x x
1( ) (ln )(1)y x x
x
1ln
du u
dx u
Example 2
1 lny x
3) (ln )b y x
2 13(ln )y x
x
23(ln )xy
x
Produ
ct Rule
Chain R
ule
![Page 108: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/108.jpg)
More Derivatives
x xde e
dx
2xy e
2( ) 2xy e
u ude e u
dx
Example 3
22 xy e
![Page 109: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/109.jpg)
Derivative of Log Functions (Base other than e)
(ln )( )x xda a a
dx
) 2xa y
(ln 2)(2 )xy
(ln )u uda a a u
dx
Example 4
2 (ln 2)xy
3) 2 xb y 3(ln 2)2 3xy
3(3ln 2)(2 )xy
![Page 110: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/110.jpg)
Derivative of Log Functions (Base other than e)
1log
(ln )a
dx
dx a x
2) loga y x
1log
(ln )a
du u
dx a u
Example 5
1
(ln 2)y
x
43) logb y x
34
14
(ln 3)y x
x
4
(ln 3)y
x
![Page 111: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/111.jpg)
Pg. 249
2 – 5, 7, 9, 21 – 23, and 30
![Page 112: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/112.jpg)
Find the First Derivative:
1)
Find the second Derivative:
2)
6xy
cos 2y x( sin 2 )(2)y x
6 ln 6xy
2sin 2y x
( 2)(cos 2 )(2)y x 4cos 2y x
![Page 113: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/113.jpg)
State Standard – 4.2 Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change.
Objective – Students will be able to solve problems involving rate of change.
![Page 114: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/114.jpg)
If we are pumping air into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. We will find an equation that relates the two quantities and then use the Chain Rule to differentiate both sides with respect to time.
![Page 115: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/115.jpg)
34
3V r
3
: 100dv cm
Givendt s
24(3 )
3
dv drr
dt dt
24dv dr
rdt dt
Example 1
3
2
1100
4 (25)
dr cm
dt s
2
1
4
dr dv
dt r dt
1
25
dr cm
dt s
Air is being pumped into a spherical balloon so that its volume increases at a rate of 100 cm3/s. How fast is the radius of the balloon increasing when the diameter is 50 cm?
: 25dr
Unknown when r cmdt
24 r24 r
![Page 116: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/116.jpg)
Steps for Related Rates Problems:
1. Draw a picture (sketch).
2. Write down known information.
3. Write down what you are looking for.
4. Write an equation to relate the variables.
5. Differentiate both sides with respect to t.
6. Evaluate.
![Page 117: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/117.jpg)
2V r h3000
min
dv L
dt
21000 ( )dv dh
rdt dt
23000 1000dh
rdt
Example 2
2
3dh
dt r
2
3000
1000
dh
dt r
How rapidly will the fluid level inside a vertical cylindrical tank drop if we pump the fluid out at the rate of 3000 L/min?
h
21000 r21000 r
?dh
dt
But since there are 1000L in a cubic meter.
21000V r h
2
3
min
m
r
The fluid level will drop at the rate of
![Page 118: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/118.jpg)
21
3V r h
3
2min
dv m
dt
21
3 2
hV h
2(3 )12
dv dhh
dt dt
Example 3
2
4dh dv
dt h dt
A water tank has the shape of an inverted circular cone with base radius 2m and height 4m. If water is being pumped into the tank at a rate of 2 m3/min, find the rate of rising water when the water is 3 m deep.
h?
dh
dt
2
4
r
h
8
9 min
dh m
dt
4r
2
2
hr
3
12V h
2
4
dv h dh
dt dt
2
4
h2
4
h
3
2
42
(3) min
dh m
dt
0.28min
m
![Page 119: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/119.jpg)
21
3V r h
3
9min
dv ft
dt
21
3 2
hV h
2(3 )12
dv dhh
dt dt
Example 4
2
4dh dv
dt h dt
Water runs into a conical tank at the rate of 9 ft3/min. The tank stands point down and has a height of 10 ft. and a base radius of 5 ft. How fast is the water level rising when the water is 6 ft. deep?
h?
dh
dt
5
10
r
h
1
min
dh ft
dt
10r
5
2
hr
3
12V h
2
4
dv h dh
dt dt
2
4
h2
4
h
3
2
49
(6) min
dh ft
dt
0.32min
ft
![Page 120: Find the derivative of the function f(x) = x 2 – 2x.](https://reader035.fdocuments.in/reader035/viewer/2022081503/56649e3f5503460f94b2fbeb/html5/thumbnails/120.jpg)
Pg. 260
1, 3 – 5, 7, 10, and 19