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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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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
 Fall '09
 BenedictGross
 Derivative, Implicit Differentiation, Logarithmic Functions

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