Derivatives of Exponential and Logarithm Functions

Derivatives of Exponential and Logarithm Functions -...

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Derivatives of Exponential and Logarithm Functions The next set of functions that we want to take a look at are exponential and logarithm functions. The most common exponential and logarithm functions in a calculus course are the natural exponential function, , and the natural logarithm function, . We will take a more general approach however and look at the general exponential and logarithm function. Exponential Functions We’ll start off by looking at the exponential function, We want to differentiate this. The power rule that we looked at a couple of sections ago won’t work as that required the exponent to be a fixed number and the base to be a variable. That is exactly the opposite from what we’ve got with this function. So, we’re going to have to start with the definition of the derivative.
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Now, the is not affected by the limit since it doesn’t have any h ’s in it and so is a constant as far as the limit is concerned. We can therefore factor this out of the limit. This gives, Now let’s notice that the limit we’ve got above is exactly the definition of the derivative of at , i.e. . Therefore, the derivative becomes, So, we are kind of stuck we need to know the derivative in order to get the derivative! There is one value of a that we can deal with at this point. Back in the Exponential Functions section of the Review chapter we stated that
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