Section 3.4

The Chain Rule

One of THE MOST POWERFUL Rules of Differentiation

The chain rule allows you to take derivatives of compositions of functions that may be hard or even impossible to differentiate with previous rules only.

Examples: not much fun, but doable with previous

techniques

Previously impossible (Unless you want to try to definition)

42 1)( xxf

322 1)( xxf

The Chain Rule

If y=f(u) is a differentiable function of u, and

u=g(x) is a differentiable function of x, then

y=f(g(x)) is a differentiable function of x and

dy/dx = dy/du du/dx

OR

)('))((')](([ xgxgfxgfdx

d

What does this mean?

When taking the derivative of a composite

function, you first take the derivative of the

outside function at the inner function and

then multiply by the derivative of the inner

function.

Common Types

1)One of the most common types of composite functions is

What the chain rule tells us to do in this case is take

the derivative with respect to the outside exponent,

and leave the inner function alone. Then, we multiply

by the derivative of the inner function (what was being

raised to the power)

nxuy )]([

Examples:

3

32

3

8

)79(

xy

ty

What about Trig functions and Exponentials?

dx

duee

dx

ddx

duuu

dx

ddx

duuu

dx

ddx

duuu

dx

d

uu )(][

tan)(sec][sec

)(sec][tan

)(cos][sin

2

dx

duuuu

dx

ddx

duuu

dx

ddx

duuu

dx

d

)cot(csc][csc

)(csc][cot

)(sin][cos

2

Examples

xy

xxy

3tan2

sinsin

Exponentials and LogarithmsLet a be a positive real number not = 1 and

let u be a differentiable function of x.

0,1

][ln udx

du

yu

dx

d

dx

du

uu

dx

d 1][ln 0,

1][ln xx

xdx

d

dx

duaaa

dx

d

aaadx

d

uu

xx

))((ln

))((ln

dx

du

uau

dx

d

xax

dx

d

a

a

)(ln

1log

)(ln

1log

More Examples:

xy

xy

xy

3

3

3

6

sec

)2ln(

And more examples…

The chain rule can be used in conjunction with both

product and quotient rules. You will need to decide

what you should do first.

2

52)(

2

52)(

2

5

5

2

x

xxg

x

xxg