chapter3(2)

# chapter3(2) - MATH1010 Chapter 3 cont 1 DIFFERENTIATION...

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MATH1010: Chapter 3 cont… 1 DIFFERENTIATION RULES cont … The Chain Rule (Section 3.5 of Stewart, pg. 217) By now, we know how to easily differentiate a function such as, say, 7 ) ( x x p = . But how do we differentiate something such as 7 2 ) 2 5 ( ) ( + = x x h ? First, notice that ) ( x h is a composite function. The Chain Rule: If f and g are both differentiable and g f F D = is the composite function defined by )) ( ( ) ( x g f x F = , then F is differentiable and F is given by the product ) ( )) ( ( ) ( x g x g f x F = In Leibniz notation, if ) ( u f y = and ) ( x g u = are both differentiable functions, then dx du du dy dx dy = So, returning to our example, if 7 2 ) 2 5 ( ) ( + = x x h , what is ) ( x h ?

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MATH1010: Chapter 3 cont… 2 Example: 8 3 4 ) 1 ( + + = x e y x Example: 1 3 4 2 + = x x y Example: x x x y cos ) 5 (sin 6 3 + = Application: The motion of a spring subject to a damping force is described by t e t f t 3 sin ) ( 2 = . What is the velocity of the spring at time t ? [Source: Modified from Calculus: Early Transcendental Functions (Single Variable, 3 rd ed) by R.T. Smith and R.B. Minton, 2007]
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chapter3(2) - MATH1010 Chapter 3 cont 1 DIFFERENTIATION...

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