# 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

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

Use the definition of the derivative.

Cancel the dark gray terms.

Divide the numerator and the denominator by *h*.

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