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!

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 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.

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 eu:

math expression


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

math expression


[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.

 

Answer

Example 2:

Find the derivative of y = ex2.


Answer

Example 3:

Find the derivative of y = sin(e3x).

Answer

Example 4:

Find the derivative of y = esin x.

 

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

LIVEMath

Normal answer:

Answer


Example 5:

Find the derivative of

math expression

Answer


Exercises

(1) Find the derivative of y = 10x2.

 

Answer


(2) Find the derivative of

math expression


Answer


(3) Find the derivative of

math expression


Answer


(4) Show that

math expression

satisfies the equation

math expression


Answer



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