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Do now as a warm-up:
Place an M&M in your mouth, but do NOT bite it. How is an M&M like a composite function?
Write your answer on the index card. You have
2.4 The Chain Rule
ex. Compose these functions to find g(f(x)) if f(x)=sinxand g(x)=
ex. If h(x) = , find two functions f and g so that
h(x) = f(g(x)).
Now, the other way
Decompose:
There is more than one correct answer
y'=So a composite function is a function with layers! ๐ฆ=๐ฅ2sin( )
๐ฆ=โ๐ฅ cos( )()4
Composite functions do not have to have a trig function as a layer.
โ7
โ๐ฅ2+1
4
Thm. The Chain RuleIf y=f(m) is a differentiable function of m and m=g(x) is a differentiable function of x, then y= f(g(x)) is a differentiable function of x and
Take the derivative of a composite function like you would unwrap a present inside a present inside a present-- one package at a time and from the outside toward the inside!
So hโ(x) =cos(m)*2x
๐ (๐ฅ )=โ2 ๐ฅ3โ2
h (๐ฅ )=sin (๐ฅ2) ๐ h๐๐ฅ
=ยฟ
๐h๐๐
=12๐
โ12
๐h๐๐
๐๐๐๐ฅ
h (๐ )=sin (๐)
๐ (๐ฅ )=๐ฅ2๐๐๐๐ฅ
=2 x
๐h๐๐
=cos (๐)
= 2x cos(x2)
๐ h๐๐ฅ
=ยฟ ๐h๐๐
๐๐๐๐ฅ
h (๐ )=โ๐
h (๐ฅ )=โโ2๐ฅ3โ2
๐๐๐๐ฅ
=โ2 ๐ฅ2
So hโ(x) ยฟ โ2 ๐ฅ2
โโ2 ๐ฅ3โ2
Thm. The General Power Rule is a special case of the Chain Rule
u'(x)
If y = and u is a differentiable function of x, then
๐ฆ=๐ ๐๐2(๐ฅ )
๐ฆ=๐ข2
๐ข=๐ ๐๐โ(๐ฅ)
*1
๐ (๐ฅ )=tan 4 (๐ฅ )
๐ข=tan (๐ฅ)
๐ข โฒ=๐ ๐๐2(๐ฅ)
๐ โฒ=4 (๐ก๐๐ยฟยฟ3(๐ฅ ))(๐ ๐๐2(๐ฅ ))ยฟ
๐๐๐๐๐๐๐๐ฆ h๐ก ๐๐
ex. If
find the derivative for all x where this function is differentiable.
,Look at a graph as a first step.
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