D_{x}(ln(x)) = 1/x 
In order to conclude that the exponential function even has an inverse function, we recall that the graph of the exponential function passes the Horizontal Line Test: no horizontal line in the xy plane touches the graph of y=e^{x} more than once. (Symbolically, this means that if e^{x}=e^{x'}, then x=x'.) The exponential function therefore has an inverse function, and the graph of this inverse function is obtained from the graph of the exponential function by reflection across y=x. (See "Horizontal Line Test" on the page "Inverse Functions.")
Basic properties of the natural logarithm are derived from properties of the exponential function and general facts about inverse functions.
(You can't take the log of a negative number!)

We can compute the derivative of the natural logarithm by using the general formula for the derivative of an inverse function. Simply set f(x)=e^{x}, so that f '(x)=e^{x} and f^{ 1}(x)=ln(x).
D_{x}(e^{ln(x)}) = D_{x}(x)
e^{ln(x)}[D_{x}(ln(x))] = 1
D_{x}(ln(x)) = 1/e^{ln(x)} = 1/x
We can now explore the derivatives of all of the exponential and logarithmic functions.
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William A. Bogley
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