Ch. 4 – More Derivatives 4.1 – Chain Rule

Ex: Find the derivative of h(x) = sin(2x 2 ). –This is not a product rule problem because it has one function inside another! It’s a composition of two functions! –To differentiate composition functions, use the CHAIN RULE! Chain Rule: –“Derivative of the outside, leave the inside alone, times the derivative of the inside” –Back to the example: Let f(x) = sinx and g(x) = 2x 2.

Ex: Find the derivative of h(x) = (3x 2 +2x – 5) 2 –This is a composition of f(x) = x 2 and g(x) = 3x 2 + 2x – 5. –First, differentiate the ( ) 2 part… –…then multiply by the derivative of what was in the ( ). Ex: Find the slope of the tangent line to f(x) = sin(1+cosx) at x=0. –First, differentiate the sin( ) part… –…then multiply by the derivative of what was in the ( ).

Ex: Find the derivative of –First, differentiate the square root… –…then multiply by the derivative of (sinx – tanx)…

Some chain rule problems have more than 2 “links”… Ex: Find the derivative of y = tan 2 (3x+1) –First, differentiate the 2 part… –…then multiply by the derivative of the tan( ) part… –…then multiply by the derivative of what was in the ( ).

Parametric Equations Usually, we represent equations using the variables x, y, and z. However, we can introduce another variable, t, as a parameter of an equation. –They look like x = f(t) and y = g(t) –The parameter, t, tells you when (as opposed to where) a point occurs on a graph –The parameter is usually restricted to specific values –Parametric equations are useful for graphing non-functions

Parametric Equations Ex: Sketch the curve given by the parametric equations x = t 2 – 4 and y = t/2, -2 ≤ t ≤ 3. –Make a table of t, x, and y, then graph! t x y ½0½13/2

Slopes of Parametrized Curves Ex: Find the slope of the curve defined by x = t 2 – 4 and y = t/2 at t = 2. –You are finding dy/dx…if you think of that as a fraction, we can rewrite it… –Slope of parametrized curves: –So for this problem, we divide the individual slopes:

Ex: Find the slope of the curve defined by at. What is the slope of the curve at the other t value, where y = ½ ? –Find what t value gives y = ½ …