3.4 Chain Rule

Prerequisite Knowledge

• Product Rule

• Quotient Rule

• Power Rule

• Special Derivatives:

Special Derivatives

€

d

dxsin x( )[ ] = cos x( )

€

d

dxcos x( )[ ] = −sin x( )

€

d

dxex

[ ] = ex

€

d

dxln x( )[ ] =

1

x

€

d

dxtan x( )[ ] = sec2 x( )

Chain Rule with Numbers

€

1

3=

1

2⋅2

3

Chain Rule With Calculus

• Why is this helpful?

• Suppose you had to differentiate:€

dy

dx=

dy

du⋅

du

dx

€

y = sin 3x 2 + 2x −1( )

Example:

€

y = sin 3x 2 + 2x −1( )

€

let u = 3x 2 + 2x −1

then du

dx= 6x + 2 and y = sin u( )

€

if y = sin u( ) then dy

du= cos u( )

€

By the Chain Rule, dy

dx=

dy

du⋅du

dx

€

So, dy

dx= 6x + 2( ) cos 3x 2 + 2x −1( )( )

Examples

€

Differentiate : y = e2x −7

€

Differentiate : y = 3x cos x 2( )

€

Determine the equation of the tangent line to

y = −2x ⋅ln x 2 + 4( ) at the origin.

Chain Rule

• Some special derivatives that come from the Chain Rule:

€

Let u be some function of x.

€

d

dxsin u( )[ ] = cos u( ) ⋅ ′ u

€

d

dxcos u( )[ ] = −sin u( ) ⋅ ′ u

€

d

dxtan u( )[ ] = sec2 u( ) ⋅ ′ u

€

d

dxln u( )[ ] =

′ u

u€

d

dxeu

[ ] = ′ u ⋅eu

Homework

• Pg. 162 # 55-92 [6]

Applications (Day 2)

• Example:

• Air is being pumped into a spherical balloon so that the radius is increasing at a rate of 2 inches per second. At what rate is the volume increasing after 3 seconds? After 10 seconds?

Example 2

• A 15 foot tall pole that was initially vertical begins to fall in such a way that the angle relative to the ground is decreasing at a rate of 3 degrees per second. At what rate is the top of the pole getting closer to the ground after 4 seconds?

Homework (2)

• Pg. 164 # 160-162

Homework (3)

• Pg. 164 # 163, 165