# Derivative of the exponential function

Categories: exponentials differentiation

In the article what is e?, we stated that:

This is an important result. It tells us that the exponential function is its own derivative. This means that we can use it to solve equations where the rate of *x* change is proportional to the value of *x*.

In this article, we will prove this result, and look at some of the consequences.

## Derivative of the exponential function

To prove the result we will start from first principles. The derivative of a function *y = f(x)* is given by:

In the case of the exponential function *e ^{x}*, this gives us:

From the rules of exponents we know that *e ^{x + h}* is

*e*so we can extract the common factor of

^{x}e^{h}*e*:

^{x}Now let's look at the definition of *e* we developed in the formula for e article:

We can substitute the value *1/h = n* is this formula:

Notice that we have changed the limit. The equation in *n* applies as *n* tends to infinity. And *n* tends to infinity, *h = 1/n* tends to zero.

We can substitute this value for *e* into the main equation:

Again from the rules of exoponents we know that (*e ^{a})^{b}* is

*e*so we can simplify this term (the power of

^{ab}*1/h*and the power of

*h*cancel each other out):

This further simplifies to:

The limit of *h/h* as *h* tends to zero is 1, so we get the final result:

## Why e?

You might still be wondering why the magic value of *e* is involved in this equation. Well if we go back to this equation:

We see that the crucial fact is that the limit term is 1:

We won't prove it here, but for this limit to be true essentially requires *e ^{x}* to have a slope of 1 at

*x = 0*.

These graphs show exponential functions with bases of 2, *e* and 5, with a dotted black line of slope 1:

Base *e* has a slope of 1 when *x* is zero. Base 2 has a slope of less than 1, base 5 has a slope of greater than 1.

## A new definition of e

We can now provide an alternative definition of *e*:

eis the base value for which the functionato the powerxhas a slope of 1 whenxis zero.

Given this new definition, it is a lot easier to understand why *e* has such special properties.

But why is this definition valid? If you recall the article formula for e, our starting point was a bank account that paid 100% per year. So £1 turned into £2 in the first year, giving a slope of £1/year.

We deliberately set the slope to 1, and the value of *e* arose out of that initial condition.

If we draw an exponential function with a slope of 1 at *x = 0*, that function will have a base of *e*. This is similar to saying that if you draw a circle of radius 1 it will have an area of *pi*.

The value *e*, like *pi*, is what it is - a strange irrational number that arises out of a simple situation.

## Integration

Since the derivative of *e ^{x}* is

*e*, it follows that the integral is also

^{x}*e*.

^{x}More precisely, the indefinite integral of *e ^{x}* is:

Where *C* is the constant of integration.

Since the function tends to zero as x tends to minus infinity, we have the following definite integral:

In other words, the area under the curve from *a*, stretching back to minus infinity, is *e to the power *a*. This is shown graphically here:

The area *A* under the curve is equal to the value of the curve

## See also

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