# Differentiation from first principles - a to the power x

Categories: differentiation calculus

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 derivative from first principles we need to be careful about the assumptions behind each method.

## Method 1 - it is a standard result

The following is a known, standard result:

This means that:

Which gives the result that:

This is the correct answer, but basing it on a standard result goes against the spirit of proof from first principles.

## Finding the limit

We would normally find a differential from first principles by evaluating the limit. From the formula above we know that:

But this is a little problematic:

- Since
*a*is positive (we stipulated this earlier) then*a*to the power*h*tends to 1 as*h*tends to 0. This means the top line of the limit equation tends to 0 as*h*tends to 0. - The bottom line of the limit equation is simply
*h*which also tends to 0 as*h*tends to 0.

So unfortunately this limit tends to 0/0, which is indeterminate, so it doesn't help us find *C*. There are a couple of ways to tackle this.

## Method 2 - L'Hopital's rule

L'Hopitals rule can be applied whenever you wish to find the limit of a function that takes the form of a quotient the top and bottom of the quotient both tend to 0:

L'Hopitals rule tells us that, in that particular case, the limit can be found from:

So if we find the derivatives of *u* and *v* we should be able to find the limit. This works well in many situations, but in this case, there is a problem. The value of *u*, the top part of the limit, is:

To differentiate that, we need to differentiate *a* to the power *x*, which is the exact problem we are trying to solve in the first place. Unfortunately this gets us nowhere.

## Maclaurin expansion

The Maclaurin expansion of *a* to the power *x* is a standard result:

We can use this with the previous formula for *Ca*:

This gives us:

We can cancel out the 1 and -1 terms:

We can cancel out the bottom *x* term by dividing every term on the top by *x*:

As *x* tends to 0, all the terms go to zero except the first *ln a* term:

This gives us the same result as before. But once again, we are relying on a standard result. And, of course, the Maclaurin expansion is calculated by differentiating the original expression, so it isn't a solution from first principles.

## The "best" solution?

Whatever way we look at this problem, we end up going around in a big (but self-consistent) circle. We need to choose a starting point and initial definition that everything else is based on.

There is no best solution, but the one that I find the simplest is based on 3 basic assumptions. First that the exponential function is its own derivative:

The second assumption is that we can use a (very basic) application of the chain rule to find the derivative of *p* times *x*:

Both of the above rules can be derived from first principles fairly easily.

And finally that the formula for changing the base of the exponential function:

Using these results, we can differentiate *a* to the *x* by expressing it as *e* to the *px* and applying the chain rule:

We can then substitute the exponential term with *a* to the *x*:

This result isn't directly derived from first principles, but it uses a couple of results that are. It is probably as close as we can get.

## See also

- Slope of a curve
- Differentiation from first principles - x²
- Second derivative and sketching curves
- Differentiation - the product rule
- Differentiation - the quotient rule
- Differentiation - the chain rule
- Differentiation - derivative of an inverse function
- Derivative of sine, geometric proof
- Differentiation - L'Hôpital's rule

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