6. Derivative of the Exponential Function

by M. Bourne

The derivative of ex is quite remarkable. The expression for the derivative is the same as the expression that we started with; that is, ex!

`(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 e2 ≈ 7.39.

Since the derivative of ex is ex, then the slope of the tangent line at x = 2 is also e2 ≈ 7.39.

We can see that it is true on the graph:

Graph derivative e^x

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

Derivative e^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 eu:

`(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 = 103x.

Example 2

Find the derivative of y = ex2.

Example 3

Find the derivative of y = sin(e3x).

Example 4

Find the derivative of y = esin x.

Example 5

Find the derivative of

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

Exercises

1. Find the derivative of y = 10x2.

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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