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

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2.4 The Chain Rule If 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 ) ( )) ( ( ) ( x g x g f x F dx du du dy dx dy

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Page 1: 2.4 The Chain Rule If 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

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

Page 2: 2.4 The Chain Rule If 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

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

Page 3: 2.4 The Chain Rule If 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

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!

Page 4: 2.4 The Chain Rule If 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

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”!)

Page 5: 2.4 The Chain Rule If 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

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

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