Finding Derivatives Using Formulas Examples

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

First write down the limit definition to be sure we remember it:

Now fill in our value a = 1 and our function f(x) = x²:

We've got it: f ' (1) = 2.

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

We have a = 1. Now we can compute the derivative:

It's good to keep those algebra skills sharp. Without them we wouldn't have been able to find that f ' (1) = -1.

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

We have a = 0.

And now we have a problem. If h is positive, the quotient evaluates to 1, so

If h is negative, the quotient evaluates to -1, so

Since the one-sided limits disagree,

Nobody said the derivative always has to exist.