The two main operations of calculus are differentiation and integration. There are many formulas that can be used in calculating the derivative of a given function. These formulas, such as the product rule, quotient rule and power rule are a great help in finding a derivative. What is obscured by using these formulas is the definition of the derivative that was used to obtain them.

The definition of the derivative is important to the subject of calculus.

Although some derivatives can be calculated directly without the use of formulas, it is good to know how to use this definition. We will see how to use the definition of a derivative, and then check our work by using another method for the derivative.

### Definition of Derivative

The derivative of a function *f* at the point *x = a* is defined as a limit. There is a conceptual reason for why this is the definition. The derivative is actually the limit of the slopes of successive secant lines which intersect the point (*a*, *f( a )* ). The slope of the line tangent at this point is the limit of the slopes of these secant lines. These slopes are formed by dividing the change in the *y* values by the change in the *x* values.

The definition of the derivative of *f* at *x = a* is as follows:

*f’ ( a )* = lim_{x → a} [*f (x ) – f( a )* ] / (*x – a*).

### Example

We know by use of the power rule that if we have the function *f( x ) = x*^{2}, then *f’ (a )* = 2 *x*.

We will now show that this is true in a different way, by using the definition of the derivative directly.

We note that if we attempt to take the limit by simply plugging in *x = a* that the difference quotient [*f (x ) – f( a )* ] / (*x – a*) will have us dividing by zero. So we cannot take this approach.

Instead we will try to algebraically manipulate the difference quotient so that we can determine the limit as *x* approaches *a*. We see that

[*f (x ) – f( a )* ] / (*x – a*) = [*x*^{2} –*a*^{2} ] / (*x – a*) .

The numerator of the difference quotient is the difference of two squares. We recall that we can factor this difference as:

*x*^{2} - *a*^{2} = (*x* - *a*) (*x* + *a*).

When we use this algebra, we see that the difference quotient becomes:

[*x*^{2} –*a*^{2} ] / (*x – a*) = [(*x* - *a*) (*x* + *a*) ] / (*x – a*) =*x* + *a*.

Now that we have cancelled the denominator of the difference quotient, it is possible to take the limit:

*f’ ( a )* = lim_{x → a} (*x* + *a*) = *a* + *a* = 2*a*.

This matches exactly with what we obtain by utilizing the power rule to calculate this derivative.

### Why Worry About the Definition?

With all of the trouble that it took with the above example, we may wonder why it is important to use the definition of the derivative. One of the main reasons is for mathematical proofs. There are places in the subject of mathematical statistics where the proper use of the definition of a derivative is a necessity.

Many times we are working with a general function for a cumulative probability function. We take the derivative of this function to obtain the probability density function.

If we are unable to use a derivative rule, such as the quotient rule or product rule, because we do not have an exact expression for our function, then the definition of a derivative will work for any function.