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!
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:
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 eu:
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 = 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.
In LiveMath, you can change the function to check your own differentiations.
Normal answer:
Example 5:
Find the derivative of
Exercises
(1) Find the derivative of y = 10x2.
(2) Find the derivative of
(3) Find the derivative of
(4) Show that
satisfies the equation
Book mark this page in Del.icio.us, Furl, Digg, StumbleUpon, whatever...
Didn't find what you are looking for? Try search:
Need a break? Play a math game. Well, they all involve math... No, really!
Home |
Sitemap |
About & Contact |
Feedback & questions |
Privacy
Users online:
19. Highest ever: 68, on 20 May 2008
Page last modified: 09 June 2007
Valid HTML 4.01
| Valid CSS