Definition of a Derivative Function

Definition of a Derivative Function

How to find the derivative of a function by using the definition of a derivative: definition, example, and its solution.

Definition

The derivative of f(x), f'(x), is the limit of [f(x + h) - f(x)]/h as h goes to 0. It is written as f'(x), y', dy/dx, and d/dx*f(x).

Recall that
f'(a) = (the limit of [f(a + h) - f(a)] / h as h → 0).

Derivative

If you put x into the a,
f'(x) = (the limit of [f(x + h) - f(x)] / h as h → 0).

Then f'(x) is the derivative function of f(x).

The derivative function is written as
f'(x), y', dy/dx, (d/dx)f(x), ... .

Example

Find f'(x) by using the definition of the derivative. f(x) = x^2 - 3x + 1

Use the definition of the derivative.

Square of a sum

Cancel the dark gray terms.

Divide the numerator and the denominator by h.

Then f'(x) = 2x - 3.

See how the slopes of the tangent lines change.

As x increases (goes to the right),
the slope increases.

That change is shown by the graph of y = f'(x).

As x increases (goes to the right),
the values of f'(x), the slope of the tangent line, increases.