1008 : The Shortcut

AP CALCULUS

Notation:

The Derivative is notated by:

( )

x

y y f x

dy d y Ddx dx

Newton L’Hopital

Leibniz

Notation:Find the rate of change of the Circumference of a Circle with respect to its Radius.

Find the rate of change of the Area of a Square with respect to the length of a Side.

Find the rate of change of the Volume of a Cylinder with respect to its Height.

Algebraic Rules

REM: A). A Constant Function

(3)ddx

( )( )

f x cf x

Algebraic Rules

B). A Power Function( )( )

nf x xf x

32

1( ) df x x y xdx x

Rewrite in exponent form!

Algebraic Rules

C). A Constant Multiplier

53 2

7 1(3 ) 4

d x y ydx x x

( )( )

nf x cxf x

Algebraic Rules

REM:

D). A Polynomial d u v

dx

3 25 2 3 4y x x x

Example: Positive Integer Powers, Multiples, Sums, and Differences

4 2 3Differentiate the polynomial 2 194

That is, find .

y x x x

dydx

Calculator: [F3] 1: d( differentiateor

[2nd ] [ 8 ] d(

d(expression,variable)

d( x^4 + 2x^2 - (3/4)x - 19 , x )

Do it all !

36 23

43y x xx

A cylindrical tank with radius 4 ft is being filled with water.

a) Write the equation for the volume of the cylinder.

b) Find the instantaneous rate of change equation of the volume with respect to the height.

c) Find the instantaneous rate of change in Volume when the height is 9 ft.

Second and Higher Order Derivatives

2

2

The derivative is called the

of with respect to . The first derivative may itself be a differentiable function

of . If so, its derivative, ,

dyy first derivativedx

y x

dy d dy d yx ydx dx dx dx

3

3

is called the of with respect to . If double prime is differentiable, its derivative,

,is called the , and so on ...

second derivative y xy y

dy d yy third derivativedx dx

Second and Higher Order Derivatives

(from superscript)

The multiple-prime notation begins to lose its usefulness after three primes.

So we use " super " to denote the th derivative of with respect to .

Do not c

n y nn y x

y

onfuse the notation with the

th power of , which is .

<< I like to use ROMAN NUMERALS through .>>

vi

n

nn y

y

yy

Example: Find all the derivatives. 5 3 23 2 5 7y x x x x

Last Update

• 08/12/10