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The Derivatives of Composite Functions We have seen many ways to differentiate combinations of polynomial functions: six): fix) 3 '(xi s-{xi fix) a '(' 3.. \ Another way that functions are often colnbined' 15 called commsition. 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 (1)6 function g ) gbc) ) function f > output f[g(x}] The new function, f(g(1t)) is called the composition function of f and g, and is written f o g . EgJ: lffbt) = 2 3x and got): 51112 +11 , then find the functions f 03 and g of and g cg. Sal: at fngix)=f{g(xii= b1 3 011x): g(f[x))= C) ss(x}= 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 # is a function of x (so that y is a composite function), then dy _ dy du du dy provided that _ and date ax exist. Eg.2: If h(x) =V2x2 +3 , find h'(x) Eg.3: If y = (x2 - x+ 2) find aEg.3: If y = (x2 - x+2) find dy dx Eg.4: If y = u +u'+ 2 where u = 1-3x2, find ay @ 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, = 30240 d.x