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Derivatives

Derivatives

At a Glance - 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.

Example 1

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


Example 2

Use the limit definition of the derivative to find f ' (1) if 


Example 3

Use the limit definition of the derivative to find f ' (0) if f(x) = |x|.


Exercise 1

Using the limit definition, what is f ' (0) if f(x) = x2?


Exercise 2

Using the limit definition, what is f ' (-1) if f(x) = 1 – x2.


Exercise 3

Using the limit definition of the derivative, what is f ' (1) if f(x) = x3?


Exercise 4

If it exists, use the limit definition to find the derivative of f ' (-1) if f(x) = x2 x.


Exercise 5

Using the limit definition to find the derivative, if it exists, what is f ' (1) if ?


Exercise 6

Using the limit definition to find the derivative, what is f ' (0) when ?


Exercise 7

Using the limit definition to find the derivative, what is f ' (0) when f(x) = x1/3?


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