Differentiation from first principles — a to the power x

In this article, we attempt to apply the first principles approach to find the derivative of a to the power x, where a is a constant value that is greater than 0. We will see that it is not quite as straightforward as we might hope.

Applying the standard formula

Differentiation from first principles uses the following standard formula to find the derivative of some function f(x):

In this case, the function is:

We can use this function in place of f(x) in the previous equation:

Using the product of powers rule we can separate the term in h:

We can then factorise the equation to completely separate the terms in x and h:

Now the term in x is independent of h, which means it doesn’t change as h tends to 0. This means we can bring it outside the limit:

Notice also that the term inside the limit doesn’t depend on x at all. This means it is a constant, that depends only on a. So we can write the derivative as:

So there we have it. The derivative of a to the power x is equal to some constant C times a to the power x. But what is this mysterious constant C? There are various ways to find it, although since our aim is to find the…