The derivative (or slope) of *f* at *a* is defined as

Since *f'*(*a*) is defined as a limit, we immediately have one strategy for finding it:

- Pick a value of
*h* that's close to *0*.

- Find the slope of the secant line between
*a* and *a* + *h*.

- Repeat (1) and (2), each time picking a smaller value of
*h*.

As we pick smaller values of *h*, we look at the slopes we're getting for the secant lines, and see if the slopes look like they're approaching some limit. If so, we can use that limit as our estimate for *f'*(*a*).

Since we're talking about a limit here, we need to do this twice: once as *h* approaches 0 from the right, and once as *h* approaches 0 from the left, to make sure the one-sided limits agree.

## Practice:

Let *f*(*x*) = *x*^{2}. 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*) = *x*^{3 }- *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*) = *x*^{2}. 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. | |