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 =

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.”

= 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

Practice these suckas! Ex. 𝑥 2 +1 100 Ex. 3 𝑥 2 +5𝑥−2 Ex. 3 𝑥 +2 3

Ex. 1 𝑥 2 + 𝑥 4 Ex. 𝑒 𝑥 2

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

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

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