2.4 the chain rule if f and g are both differentiable and f is the composite function defined by...

2.4 The Chain RuleIf f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F′ is given by the product

In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then

)())(()( xgxgfxF

dx

du

du

dy

dx

dy

Example:

2sin 4f g x x

2sin 4y x

siny u 2 4u x

cosdy

udu

2du

xdx

dy dy du

dx du dx

cos 2dy

u xdx

2cos 4 2dy

x xdx

2sin 4y x

2 2cos 4 4d

y x xdx

2cos 4 2y x x

A faster way to write the solution:

Differentiate the outer function...

…then the inner function

Another example:

2cos 3d

xdx

2cos 3

dx

dx

2 cos 3 cos 3d

x xdx

derivative of theouter function

derivative of theinner function

It looks like we need to use the chain rule again!

Another example:

2cos 3d

xdx

2cos 3

dx

dx

2 cos 3 cos 3d

x xdx

2cos 3 sin 3 3d

x x xdx

2cos 3 sin 3 3x x

6cos 3 sin 3x x

The chain rule can be used more than once.

(That’s what makes the “chain” in the “chain rule”!)

The Power Rule combined with the Chain Rule

1n nd duu nu

dx dx

)()()( 1 xgxgnxgdx

d nn

If n is any real number and u=g(x) is differentiable, then

Alternatively,

Example:

492502

2492502

31002350

33503

xxxx

xdx

dxx

dx

d