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 + ./0 1 Rate yourself on how well you understood this lesson. I understand I don’t get I sort of I understand most of it but I I got it! it at all get it it pretty well need more practice 1 2 3 4 5 What do you still need to work on? .