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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! 𝑦=𝑥2sin( )
𝑦=√𝑥 cos( )()4
Composite functions do not have to have a trig function as a layer.
−7
√𝑥2+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 h’(x) =cos(m)*2x
𝑚 (𝑥 )=−2 𝑥3−2
h (𝑥 )=sin (𝑥2) 𝑑 h𝑑𝑥
=¿
𝑑h𝑑𝑚
=12𝑚
−12
𝑑h𝑑𝑚
𝑑𝑚𝑑𝑥
h (𝑚 )=sin (𝑚)
𝑚 (𝑥 )=𝑥2𝑑𝑚𝑑𝑥
=2 x
𝑑h𝑑𝑚
=cos (𝑚)
= 2x cos(x2)
𝑑 h𝑑𝑥
=¿ 𝑑h𝑑𝑚
𝑑𝑚𝑑𝑥
h (𝑚 )=√𝑚
h (𝑥 )=√−2𝑥3−2
𝑑𝑚𝑑𝑥
=−2 𝑥2
So h’(x) ¿ −2 𝑥2
√−2 𝑥3−2
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
𝑦=𝑠𝑖𝑛2(𝑥 )
𝑦=𝑢2
𝑢=𝑠𝑖𝑛❑(𝑥)
*1
ex. If
find the derivative for all x where this function is differentiable.
,Look at a graph as a first step.