The Chain (Saw) Rule Lesson 3.4

The Chain Rule According to Mrs. Armstrong … “Pull the chain and the light comes on!”

Introduction Sludge Falls CO 2 is changing at rate of 0.02 ppm for each person Population growing at rate of 1000 people/yr We seek rate of increasing pollution with respect to time (0.02 ppm/prsn)(1000 people/yr) = 20 ppm/yr

A Composite Function The level of pollution L is a function of the population P, which is itself a function of time t. L = f(P(t)) Then L’ … is Rate of Change of L with respect to t = Rate of change of L with respect to P Rate of change of P with respect to t

A Composite Function In Leibniz notation: Result in pollution as a function of time Pollution as a function of the population Population as a function of time

The Chain Rule Given y = f(u) is a differentiable function of u u is also a differentiable function … of x Then y = f(u(x)) Then

Example Given: y = (6x 3 – 4x + 7) 3 Then u(x) = 6x 3 – 4x + 7 and f(u) = u 3 Thus f’(x) = 3(6x 3 – 4x + 7) 2 (18x 2 – 4)

Example Given g(u) = u 5 u(x) = 3x + 1 Then g’(u) = ?? u’(x) = ?? f(x) = (3x + 1) 5 f’(x) = ??

Example Find equation of tangent line to at (2,3)

Example Try Which is the u(x), the “inner” function? Which is the f(u), the “outer” function? What is u’(x), f’(u) ??

Example Try with multiple levels of nested functions

Example Try in combination with the product rule

Assignment Part A Lesson 3.4A Page 161 Exercises 1 – 49 Odd

Derivative for ln x Graph the difference function for ln x This has a familiar look, pardner!

Derivatives for Bases Other Than e View Geogebra Demo View Geogebra Demo View Geogebra Demo View Geogebra Demo

One More Try Determine the derivative What is the equation of the tangent line of f(x) when x = 1?

Assignment Part B Lesson 3.4B Page 162 Exercises 51 – 127 EOO