Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2...

The Chain Rule

Section 3.6c

Suppose that functions f and g and their derivatives have thefollowing values at x = 2 and x = 3.

x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5

Evaluate the derivatives with respect to x of the followingcombinations at the given value of x.

2 f x(a) at x = 2

2d

f xdx

2 f x 2 2f 1 2

23 3

At x = 2:


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


f x g x(b) at x = 3

df x g x

dx

f x g x 3 3f g 2 5 At x = 3:


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


f x g x(c) at x = 3

df x g x

dx f x g x g x f x

3 3 3 3f g g f 3 5 4 2 At x = 3:

15 8


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


f x g x(d) at x = 2

2 2d

f gdx

2

2 2 2 2

2

g f f g

g

22 1 3 8 3

2

74 3 37

4 6


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


f g x(e) at x = 2

2df g

dx 2 2f g g 2 2f g

13 1

3


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


f x(f) at x = 2

2df

dx 12

2 2

df

dxf

2

2 2

f

f

1 3

2 8

1

12 2


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


21 g x(g) at x = 3

23

dg

dx

3

2 3 3d

g gdx

3

2 3

3

g

g

32 5

4

5

32


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


2 2f x g x(h) at x = 2

2 22 2d

f gdx

2 2

12 2 2 2 2 2

2 2 2

d df f g g

dx dxf g


x2

3

f x

8

3

g x

2

–4

f x

1/3

2

g x

–3

5


2 2f x g x(h) at x = 2

2 2

2 2 2 2

2 2

f f g g

f g

2 2

8 1 3 2 3

8 2

10 3

68

10 3

2 17

5

3 17

Slopes of Parametrized Curves

dy dy dx

dt dx dt

Usually, we write this in a different form…

A parametrized curve (x(t), y(t)) is differentiable at t if x and yare differentiable at t. At a point on a differentiable parametrizedcurve where y is also a differentiable function of x, thederivatives dy/dt, dx/dt, and dy/dx are related by the Chain Rule:

0dx dt dy dy dt

dx dx dt

If all three derivatives exist and ,

Practice Problems

sin 2x t

Find the equation of the line tangent to the curve at the pointdefined by the given value of t.

cos 2y tFind the three derivatives:

1 6t

cos 2 2dx d

t tdt dt

2 cos 2 t

sin 2 2dy d

t tdt dt

2 sin 2 t

dy dy dt

dx dx dt 2 sin 2

2 cos 2

t

t

tan 2 t

Practice Problems

sin 2x t


cos 2y t

The line passes through:

1 6t

2 2sin ,cos

6 6

3 1,

2 2

And has slope:

2tan

6

3

Equation of the tangent line:

1 33

2 2y x

3 2y x

Practice Problems

22 3x t


4y t

Derivatives:

1t

4dx

tdt

34dy

tdt

dy dy dt

dx dx dt

324

4

tt

t

Point: 2 42 1 3, 1 5,1 Slope: 21 1

Equation of the tangent line:

1 1 5y x 4y x

Suppose that functions f and g and their derivatives have the following values at x = 2 and x = 3. 2...

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