1. Warmup: Decompose the following functions
Lesson: ____
Section: 3.4
The Chain Rule
Warmup: Decompose the following functions
h(x) = f(g(x))
Outer Function
Inner Function
Ex. ℎ 𝑥 = 2𝑥+1
Ex. ℎ 𝑥 = 𝑒 𝑥
Ex. ℎ 𝑥 = 1 (𝑥 3 +2𝑥) 2
u = g(x) = 2x + 1
𝒇 𝒖 = 𝒖
u = g(x) =
𝒇 𝒖 =
u = g(x) =
𝒇 𝒖 =

2. If f and g are differentiable functions, then
𝐼𝑓 𝑢=𝑔 𝑥 , 𝑦=𝑓 𝑢 , 𝑡ℎ𝑒𝑛 𝑦=𝑓(𝑔 𝑥 )
A small change in x will generate a small change in u which, in turn, generates a small change in y x u y
g(x)
f(u)
f(g(x))
𝑑𝑦 𝑑𝑥 = 𝑑𝑦 𝑑𝑢 ∙ 𝑑𝑢 𝑑𝑥
𝑑 𝑑𝑥 𝑓(𝑔 𝑥 )
𝑓′ 𝑢
g′ 𝑥
The Chain Rule
If f and g are differentiable functions, then
𝑑 𝑑𝑥 𝑓(𝑔 𝑥 )= 𝑓 ′ (𝑔 𝑥 )∙𝑔′(𝑥)
“The derivative of a composite function is… The derivative of the outer function(evaluated at the inner) multiplied by the derivative of the inner.”

3. = 𝑑𝐿 𝑑𝑇 = 𝑑𝑇 𝑑𝑡 = 𝑑𝐿 𝑑𝑡 = 𝑑𝐿 𝑑𝑇 ∙ 𝑑𝑇 𝑑𝑡 = 2 𝑐𝑚 ℃ 3 ℃ ℎ𝑟 = 6 𝑐𝑚 ℎ𝑟
Suppose the length L, in cm of a steel bar depends on the air temperature, T in C and Temperature depends on time, t, measured in hours.
If the length of the bar increases by 2 cm for every degree increase in temperature, and the temperature increases by 3 C per hour, how fast is the length of the bar increasing? What are the units? t T L
L depends on T and T depends on t
Rate L is increasing with respect to T
= 𝑑𝐿 𝑑𝑇
=2 𝑐𝑚 ℃
Rate T is increasing with respect to t
= 𝑑𝑇 𝑑𝑡
=3 °𝐶 ℎ𝑟
Rate L is increasing at with respect to t
= 𝑑𝐿 𝑑𝑡 = 𝑑𝐿 𝑑𝑇 ∙ 𝑑𝑇 𝑑𝑡
= 2 𝑐𝑚 ℃ 3 ℃ ℎ𝑟
= 6 𝑐𝑚 ℎ𝑟

4. Practice these suckas!
Ex. 𝑥
Ex 𝑥 2 +5𝑥−2
Ex 𝑥 +2 3

5. Ex. 1 𝑥 2 + 𝑥 4
Ex. 𝑒 𝑥 2

6. Reviewing the concept…
Ex. y is a function of 𝑥. 𝑦=𝑓 𝑥
𝑑 𝑑𝑥 𝑥=
𝑑 𝑑𝑥 𝑥 2 =
𝑑 𝑑𝑥 𝑦 2 = 1
Extending the idea a little further… If 𝑦=𝑓 𝑥 .
𝑑 𝑑𝑥 3𝑥𝑦 2 =
3∙ 𝑑 𝑑𝑥 𝑥 𝑦 2 2𝑥
3∙ 1 𝑦 2 + 𝑥 ( )
2𝑦∙𝑦′
2𝑦∙𝑦′
3 𝑦 2 +6𝑥𝑦∙𝑦′
You fools want more practice?
Try #4, 28, 36, 50, 54, 56 or even 46
Hint: Think of this as
𝒅 𝒅𝒙 𝒇 𝒙 𝟐

7. Ex. Try d dx x 2 +1 100 again without declaring new variables.

8. Try slaying a dragon! Apply the chain rule multiple times…
Ex. d dx 𝑒 − 𝑥 7 +5