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!

math expression

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 e^x is e^x, 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.

Deriv 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 e^u:

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

[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.

Answer

Example 2:

Find the derivative of y = e^x².

Answer

Example 3:

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

Answer

Example 4:

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

In LiveMath, you can change the function to check your own differentiations.

LIVEMath

Normal answer:

Answer

Example 5:

Find the derivative of

Answer

Exercises

(1) Find the derivative of y = 10^x².

Answer

(2) Find the derivative of

Answer

(3) Find the derivative of

Answer

(4) Show that

satisfies the equation

Answer

5. Derivative of the Logarithmic Function

7. Applications: Derivatives of Logarithmic and Exponential Functions

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