Let f(x) = x². Estimate the derivative of f at 1.

We already did this problem! If we go back and look at this exercise, we evaluated the slope of the secant line from a = 1 to a + h for h = -0.1,-0.2,-0.3,-0.4, 0.4,0.3,0.2,0.1 and found that as h approached 0, the slope of the secant line approached 2. This means our best guess is that f'(1) = 2.

Let f(x) = x³ - x. Estimate the derivative of f at 0.

We have a = 0. Find the slopes of the secant lines from x = a to x = a + h for some values of h approaching 0 from either side, and see if the slopes approach a limit. First let h approach 0 from the right.

Looks like

Now see what happens as h approaches 0 from the left.

Yup, we also have

This means our best estimate is

f'(0) = -1.

There's nothing special about having h change by 0.1 each time.

Let f(x) = x². Estimate the derivative of f at 0.

First have h approach 0 from the right.

Now we have h approach 0 from the left.

The slopes of the secant lines approach 0 whether h approaches 0 from the left or right. We conclude that the derivative of f at 0 is 0:

f'(0) = 0.

Let . Estimate the derivative of f at 1. 

Hint: we won't have pretty numbers.

First, have h approach 0 from the right.

Now have h approach 0 from the left.

The slopes of the secant lines appear to approach 0 whether h approaches 0 from the left or right. We estimate that

f'(0) = -1.

Since the derivative is defined as a limit, there will be occasions where the derivative doesn't exist.

Find the derivative of f(x) = |x| at 0.

Do what we've been doing. First let h approach 0 from the right.

We see that as h approaches 0 from the right, we always find a slope of 1:

What happens as h approaches 0 from the left?

We see that as h approaches 0 from the left, we find slopes of -1:

Since the one-sided limits do not agree,

  does not exist.