Summary

The logarithmic function has a really interesting property (which mirrors the really interesting property of the function ): it is the only non-power function that has as its derivative a power. In fact,

So it's easy to tell the slope of the tangent line to the function at a particular value of x: it's 1/x.

This is actually pretty easy to derive: consider , and use the chain rule, the derivative of the exponential function, and the inverse property of these two functions to write

Differentiating both sides, we find that

or

Don't worry too much about other bases. You can always write a logarithmic function in terms of the natural logarithm:

If you have a composition involving logs, then you can make use of this special form of the chain rule: if you have , then

#1, p. 260

#8, p. 260

#16, p. 260

#48, p. 261