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Unformatted text preview: DERIVATIVES OF LOGARITHMIC FUNCTIONS We can use the method of implicit differentiation to find derivatives of logarith- mic functions as well as inverse trigonometric functions. The proof is similar to the inverse trigonometric function case. Let y = log a x . Then a y = x . Differentiating this implicitly with respect to x , we get a y (ln a ) y = 1 . Thus, y = 1 a y ln a = 1 x ln a . When a = e , ln a = 1, and we get d dx (ln x ) = 1 x . We can take another approach to explain this. For any n 6 =- 1, x n has an anti- derivative. When n =- 1, however, we had not known its antiderivative, because if we use the power rule it suggests that an antiderivative of x- 1 is a multiple of x , or a constant. Since the derivative of a constant is zero, we have something wrong here. Thus, we need to find a totally different function as an antiderivative of x- 1 , and it is a natural logarithm. In fact, some textbooks use this for the definition of e ....
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