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# Derivatives

# Slope at One Point?

The slope of the function *f* at the point (*a*,*f*(*a*)) is the limit of the slope of the secant line between *x* = *a* and *x* = *a* + *h *as *h* gets closer to *0*. Whew, that's a mouthful.

From the formula for the slope of this secant line, we find another formula:

"The slope of *f* at *a*" is also called the **derivative** of *f* at *a* and is written *f'*(*a*). This nice formula is known as the **limit definition of the derivative**, and we write it again here, with correct notation:

Visually, we move *h* closer and closer to 0 (equivalently, move *a + h* closer and closer to *a*) and watch what happens to the secant line, as in this animation.

If we could keep going until *h* reached *0* (equivalently, until *a* + *h* reached *a*) we would find a line that, instead of passing through the graph twice, would hit the graph at one spot and bounce off, as shown in this picture:

The slope of the function *f* at the single point *x* = *a* is the slope of this line, also called the tangent line.

Two important things to remember:

- The
**slope**of*f*at*a*and the**derivative**of*f*at*a*are the same thing.

- Since the derivative of
*f*at*a*is a limit, the derivative won't always exist

Another phrase for *f'*(*a*) is the **instantaneous rate of change of f **with respect to

*when*

*x**x = a*.