From 11:00PM PDT on Friday, July 1 until 5:00AM PDT on Saturday, July 2, the Shmoop engineering elves will be making tweaks and improvements to the site. That means Shmoop will be unavailable for use during that time. Thanks for your patience!
We have changed our privacy policy. In addition, we use cookies on our website for various purposes. By continuing on our website, you consent to our use of cookies. You can learn about our practices by reading our privacy policy.
© 2016 Shmoop University, Inc. All rights reserved.
Derivatives

Derivatives

Finding Derivatives Using Formulas

In this section we need to find derivatives "analytically," also known as "using the limit definition."

Be Careful: "Find the derivative using the limit definition" does not mean estimating the derivative like we did earlier . There, we were estimating. Now, we'll be exact.

Sample Problem

Let f(x) = x2 + 1. What is f ' (1) using the limit definition of the derivative?

We have a = 1 and f(x) = x2 + 1, so we find

Be careful here. Make sure to plug in (1 + h) to f, and don't forget about the extra + 1 at the end of the function f:

Since we evaluated the limit properly, rather than estimating it, we can conclude that f ' (1) = 2.

Sample Problem

Let f(x) = x2x. Use the limit definition of the derivative to find f '(2).

We have a = 2.

Here's another place to be careful. It's possible to lose x's and h's in confusing spots like this, evaluate f(2 + h) and f(2) separately.

Putting things back in,

We can conclude that f ' (2) = 3.

If derivatives take this long, how does anyone finish their homework in time to watch The Late Show? Why doesn't anyone's arm ever fall off during a calc exam? In answer to these questions, yes, there are easier ways of calculating derivatives. However, "the limit definition of the derivative" is important, and it will be on the test.

People who Shmooped this also Shmooped...

Advertisement