Warm Up:

h(x) is a composite function of f(x) and g(x). Find f(x) and g(x).

1. 2. 42)( xxh xxh 2sin)(

THE CHAIN RULE

Objective: To use the chain rule to find the derivative of composite functions.

What if we wanted to take the derivative of the following:

1. y=(2x+5)2

2. y=(2x+5)3

3. y=(2x+5)10

If we had y= 6x -10, dy/dx = 6. EASY BREEZY

What if we wrote y as a composite function?y = 2(3x -5)Outer function: Inner function: y = 2u u = 3x-5 2du

dy 3dx

du

632 dx

du

du

dy

dx

dy

The Chain Rule (used for composite functions)If y is a function of u, y= f(u), and u is a function of x, u=g(x),

then y = f(u) = f(g(x)) and :

If we have a composite function:Derivative of = derivative of X derivative of function outer function inner function

(You may have composites within composites so may have to repeat)

dx

du

du

dy

dx

dy

If y=f (g(x)), then y’=f ’ (g(x)) g’(x)∙

Derivative of outer function, times derivative of inner. Take a look:y=(2x-1)2

Examples: Find the derivative.

1. y = (3x +4 )3

2. 13 xy

Find the derivative.

1. y= sin(x2 + x)

2. y= (x3 +2x) -1

3. y= cot(x2)

223 164 xxy

Chain rule with products and quotients. Good times, good times….

1. y = 3x(x3 +2x2)3

2. y =

3. y = sin 2x cos2x

432 xx

Quotient rule…do we need it?? You decide.

Find the derivative: xx

xy

3

22

43

2

23

4)(

t

tttp

2

2 1

56

x

xy

Find the equation of the tangent line to the graph of at x = 3. 16)( 2 xxf

Determine the point(s) at which the graph of has a horizontal tangent. 12

)(

x

xxf

Use the table of values to find the derivative.

x 1 2 3 4

f(x) 2 4 1 3

f ‘ (x) -6 -7 -8 -9

g(x) 2 3 4 1

g ‘ (x) 2/7 3/7 4/7 5/7

FIND at x = 2. ))(( xgfDx

Repeated Use of Chain Rule: Take derivative of outer function and work your way in until you have no more composites.

f(g(h(x)))1. f(x) = cos2(3x)

Find the derivative.

1. f(x) = tan(5 – sin2t)

2. y = 4 3 55 x