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lecure16 - 3.7 Derivatives of Inverse Functions Derivatives...

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3.7 Derivatives of Inverse Functions Derivatives of Inverses of Differential Functions What is the inverse function of f ( x ) = x 2 + 1? 1
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THEOREM 4 - The Derivative Rule for Inverses If f has an interval I as domain and f ( x ) exists and is never zero on I , then f - 1 is differentiable at every point in its domain. The value of ( f - 1 ) at a point b in the domain of f - 1 is the reciprocal of the value of f at the point a = f - 1 ( b ): Example 1 f ( x ) = x 2 , find ( f - 1 ) ( x ) ( derivative of the inverse ) 2
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Derivative of the Natural Logarithmic Function Example Solve for x : 17 = 19 x What is the inverse of f ( x ) = e x ? ( f - 1 ) ( x ) = d dx (ln | x | ) = If u is a differentiable function of x with u > 0, we get ( applying the Chain Rule ) d dx (ln u ) = Example 3 1. d dx ln 2 x 3
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2. d dx ln(
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