For **average rate of change from x = a to x = a + h** is:

[f(a + h) - f(a)]/h, h ≠ 0

For **instantaneous rate of change at x = a** is:

limh → 0[f(a + h) - f(a)]/h

if the limit exists.

Given **slope of the graph** at the point *(a, f(a))* is given by:

limh → 0[f(a + h) - f(a)]/h

provided the limit exists. In this case, the **tangent line** to the graph is the line through *(a, f(a))* with slope given by this limit.

Drag the sliders to see the slope of the secant line for different values of *x* and *h*.

- For y = f(x), we define the
**derivative of**, denoted*f*at*x*, by:*f'(x)*

if the limit exists.f'(x) = limh → 0 [f(x + h) - f(x)]/h

- Other representations for the derivative of
*f*at*c*are*y'*and*dy/dx*. - If f'(x) exists for each
*x*in the open interval (*a, b*), then*f*is said to be**differentiable**over (*a, b*). - If this limit does not exist at
*x = a*, we say that the function*f*is**nondifferentiable at**, or*x = a*.*f'(a)*does not exist *f'(a)*does not exist if the graph of*f*has:- a hole or a break at
*x = a*. - a sharp corner at
*x = a*. - a vertical tangent at
*x = a*.

- a hole or a break at

The derivative of a function *f* is a new function *f'*. The domain of *f'* is a subset of the domain of *f*. The derivative has various applications and interpretations, including the following:

*Slope of the tangent line*. For each*x*in the domain of*f'*,*f'(x)*is the slope of the line tangent to the graph of*f*at the point (*x, f(x)*).*Instantaneous rate of change*. For each*x*in the domain of*f'*,*f'(x)*is the instantaneous rate of change of y = f(x) with respect to*x*.*Velocity*. If*f(x)*is the position of a moving object at time*x*, then v = f'(x) is the velocity of the object at that time.