# 6. Derivative of the Exponential Function

by M. Bourne

The derivative of *e ^{x} *is quite remarkable. The expression for the derivative is the same as the expression that we started with; that is,

*e*!

^{x}`(d(e^x))/(dx)=e^x`

**What does this mean? **It means the slope is the same as the function value (the *y*-value) for all points on the graph.

**Example: **Let's take the example when *x* = 2. At this point, the *y*-value is *e*^{2} ≈ 7.39.

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

*e*, then the slope of the tangent line at

^{x}*x*= 2 is also

*e*

^{2}≈ 7.39.

We can see that it is true on the graph:

Let's now see if it is true at some other values of *x*.

We can see that at *x* = 4, the *y*-value is 54.6 and the slope of the tangent (in red) is also 54.6.

At *x* = 5, the *y*-value is 148.4, as is the value of the derivative and the slope of the tangent (in green).

## Other Formulas for Derivatives of Exponential Functions

If *u* is a function of *x*, we can obtain the derivative of an expression in the form *e ^{u}*:

`(d(e^u))/(dx)=e^u(du)/(dx)`

If we have an exponential function with some base *b*, we have the following
derivative:

`(d(b^u))/(dx)=b^u\ ln\ b(du)/(dx)`

[These formulas are derived using first principles concepts. See the chapter on Exponential and Logarithmic Functions if you need a refresher on exponential functions before starting this section.]

### Example 1

Find the derivative
of *y* = 10^{3x}.

### Example 2

Find the derivative
of *y* = *e*^{x2}.

### Example 3

Find the derivative
of *y* = sin(*e*^{3x}).

### Example 4

Find the derivative
of *y* = *e*^{sin x}.

### Example 5

Find the derivative of

`y=(ln\ 2x)/(e^(2x)+2`

## Exercises

1. Find the derivative of *y* = 10^{x2}.

2. Find the derivative of

`y=cos\ 2x(e^(x^2-1))`

3. Find the derivative of

`y=(2e^(x^2)+x^2)^3`

4. Show that

`y=e^(-x)sin\ x`

satisfies the equation

`(d^2y)/(dx^2)+2(dy)/(dx)+2y=0`

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