Section 2.4: The Derivative

Average Rate of Change

For y = f(x), the average rate of change from x = a to x = a + h is:

[f(a + h) - f(a)]/h, h ≠ 0

Instantaneous Rate of Change

For y = f(x), the instantaneous rate of change at x = a is:

limh → 0[f(a + h) - f(a)]/h

if the limit exists.

Slope of a Graph and Tangent Line

Given y = f(x), the 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.

The Derivative

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

  f'(x) = limh → 0 [f(x + h) - f(x)]/h

  if the limit exists.
• 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 x = a, or 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.

Interpretations of the Derivative

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:

1. 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)).
2. 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.
3. 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.