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Math_137_Winter_2010_Week_6_Notes

Math_137_Winter_2010_Week_6_Notes - Math 137 Week 6 Notes...

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Math 137 Week 6 Notes – Dr. Paula Smith 3.4 The chain rule gives the derivative of a composition of two functions. If g is differentiable at x and f is differentiable at g(x), then f(g(x)) is differentiable at x, and can be evaluated as dx dg dg df x g x g f x g f = = ) ( * )) ( ( ] )) ( ( [ To get used to this formula, write down f(x), f (x), g(x), g (x), and f (g(x)). Then the answer is f (g(x)) * g (x). If a function is a composition of three functions f, g and h, we can easily extend the chain rule: dx dh dh dg dg df x h x h g x h g f x h g f = = ) ( * )) ( ( * ))) ( ( ( ] ))) (( ( ( [ 3.5 When y = f(x), we say y is explicitly a function of x. Here, the slope of function y = f(x) is dx dy y = . But we can extend this idea to determine the slope of a curve expressed in both x and y – where y is implicitly (one or more) function(s) of x, but y cannot be isolated on one side of the equation sign. Given such an equation, take

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Math_137_Winter_2010_Week_6_Notes - Math 137 Week 6 Notes...

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