AP Calculus BC 3.6 Chain Rule Objective: able to differentiate composite functions using the Chain Rule (the most widely used differentiation rule); to find slopes of parameterized curves.

1. Let sin and 1. Find , , ∘ .

2. The function sin 1 is the composition of the functions f & g. Let sin & 1. How are the derivatives of these three related?

The Chain Rule (Newton Notation) = ( ) If f is differentiable at the point ugxg , and is

differentiable at x , then the composite function ()()()fgxfgxa = () is differentiable at x , and

′ ()()()fga x= fgx′() ⋅ gx ′ ()

‘Outside – Inside’ Rule

It sometimes helps to think about the Ch ain Rule this way: dy If yfgx=()() , then = fgxgx′()() ⋅ ′ () . dx In words, differentiate the "outside" fu nction f and evaluate it at the

"inside" function g ( x ) left alone; then multiply by the derivative of the

"inside" function.

The Chain Rule (Leibniz Notation)

3. An object moves along the x-axis so that its position at any time ≥ 0 is given by the function . Find the velocity of the object as a function of time. cos − 3

4. Find . √100 + 8

Slopes of Parametrized Curves A parametrized curve ( x( t) , y( t)) is differentiable at t if

x and y are differentiable at t .

dx If all three derivatives exist and ≠ 0, dt

dy dy = dt dx dx

dt

5. Find the equation of the line tangent to the curve defined parametrically by = 2cos and at the point where . = 2sin =

Power Chain Rule

If f is a differentiable function of u , and u is a differentiable function of x , then substituting yfu= () into the Chain Rule

dy dy du formula = ⋅ leads to the formula dx du dx

d du fu()()= f′ u dx dx

More specifically, if n is an integer and , the Power Rules tell us that ′ . So when u is a differentiable function of x, then the Power Chain Rule gives us :

6. Given 1, find ′.

Repeated use of the Chain Rule 7. Find if . = 1 + 7

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