Unit 2 - Activity 5 n The Derivatives of Composite Functions { We have seen many ways to differentiate combinations of polynomial functions: Another way that functions are often combined 15 called composition. In this case, one function 15 substituted for another. You can think of a composite function as an input- output diagram, when the output of one function is used as the input for a second. input (x) -> function g > g(x) > function f > output f(g(x)) The new function, f(g(x)) is called the composition function of f and g, and is written f o g . Eg.1: If f(x) : 2 3X and g(x) = 5x2 + x , then find the functions f o g and g o f and g o g . Sol: a) f 0 sh): f(s(X)l= b) g of(x)= g(f(x))= c) g 0 06) = To find the derivative of a composite function h(x) : f (g(x)) we use the chain rule. MCV4U The Chain Rule If f and g are functions having derivatives, then the composite function h(x) = f (g(x)) has a derivative given by: Work from the outside to the inside. In words we say the derivative of the outer function evaluated at the inner function times the derivative of the inner function. In Liebniz Notation: If y is a function of u and u is a function of x (so that y is a composite function), then dy = dudx provided that , and 7 exist. Eg.2: If h(x) =V2x2 +3 , find h'(x) Eg.3: If y = (x2 - x+ 2) find dy dx Eg.4: If y = ul tus + 2 where u = 1-3x2, find - @ x=1. dx Sol: Using the Leibniz Notation, dy _ dy du =(10u + 5u4). (-6x) dx du dx It is not necessary to write this expression entirely in terms of x. Note that when x = 1 we have u = -2. Therefore, dy = 30240 dx Homework: p. 105 # 1 def, 4, 8acf, 10, 13abd, 16, 17b