lecture14 - Repeated Use of the Chain Rule We sometimes...

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3.5 The Chain Rule Derivative of a Composite Function Example 1 y = (3 x 2 + 1) 2 THEOREM 3 - The Chain Rule If f ( u ) is differentiable at the point u = g ( x ) and g ( x ) is differentiable at x , then the composite function is differentiable at x and Example 2 An object moves along the x -axis so that its position at any time t 0 is given by x ( t ) = cos( t 2 + 1). Find the velocity of the object as a function of t . 1
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”Outside-Inside” Rule If y = f ( g ( x )), then Example 3 sin( x 2 + e x ) Example 4 y = e cos x d dx e u = e u du dx . 2
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Unformatted text preview: Repeated Use of the Chain Rule We sometimes have to use the Chain Rule two or more times to nd a deriva-tive. Example 5 g ( t ) = tan(5-sin (2 t )) The Chain Rule with Powers of a Function Let u be a dierentiable function of x , then d dx u n = This is called the Power Chain Rule . Example 6 1. d dx (5 x 3-x 4 ) 7 3 2. d dx ( 1 3 x-2 ) 3. d dx (sin 5 x ) 4. d dx ( e 3 x +1 ) 4...
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lecture14 - Repeated Use of the Chain Rule We sometimes...

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