# 1.2 Combining differentiable functions

It would be tedious if we had to use the definition of the derivative every time we needed to differentiate a function. Fortunately, once the derivatives of simple functions like and are known, we can find the derivatives of other more complicated functions by applying the following theorem.

## Theorem 3 Combination Rules for Differentiation

Let and be complex functions with domains and , respectively, and let be a limit point of . If and are differentiable at , then

1. Sum Rule is differentiable at , and

2. Multiple Rule is differentiable at , for , and

3. Product Rule is differentiable at , and

4. Quotient Rule is differentiable at (provided that , and

We remark that if the domains and in Theorem 3 are regions, then every point of is a limit point of and of .

In addition to these rules, there is a corollary to Theorem 3, known as the Reciprocal Rule, which is a special case of the Quotient Rule.

## Corollary Reciprocal Rule for Differentiation

Let be a function that is differentiable at . If , then is differentiable at , and

The proof of the Combination Rules for differentiation uses the Combination Rules for limits of functions. In the next example we illustrate the method by proving the Product Rule for differentiation. We use the Sum, Product and Multiple Rules for limits of functions, and we also use the fact that if a function is differentiable at , then it is continuous at , so .

## Example 2

Prove the Product Rule for differentiation.

## Solution

Let . Then

The proofs of the other Combination Rules are similar. We ask you to prove the Sum and Multiple Rules in Exercise 4, and the Quotient Rule later in Exercise 12.

## Exercise 4

Prove the following rules for differentiation.

1. Sum Rule

2. Multiple Rule

### Answer

1. Let . Then

2. Let , for . Then

The Combination Rules enable us to differentiate any polynomial or rational function. (Recall that a rational function is the quotient of two polynomial functions.)

For example, since the function is entire with derivative , we can use the Product Rule repeatedly to show that the function

is entire, and that its derivative is

(This result can be proved formally using the Principle of Mathematical Induction.) Next, we can use this fact, together with the Sum and Multiple Rules, to prove that any polynomial function is entire, and that its derivative is obtained by differentiating the polynomial function term by term. For example,

In general, we have the following corollary to Theorem 3.

## Corollary Differentiating Polynomial Functions

Let be the polynomial function

where and . Then is entire with derivative

Since a rational function is a quotient of two polynomial functions, it follows from the corollary on differentiating polynomial functions and the Quotient Rule that a rational function is differentiable at all points where its denominator is non-zero; that is, at all points of its domain.

## Example 3

Find the derivative of

and specify its domain.

## Solution

By the corollary on differentiating polynomial functions, the derivative of is

and the derivative of is

Provided that is non-zero, we can apply the Quotient Rule to obtain

Since is non-zero everywhere apart from and , it follows that the domain of is .

## Exercise 5

Find the derivative of each of the following functions. In each case specify the domain of the derivative.

### Answer

1. By the corollary on differentiating polynomial functions, we have

2. By the Quotient Rule,

   Now, if and only if , so the domain of is

So, any rational function is differentiable on the whole of its domain. What is more, this domain must be a region because it is obtained by removing a finite number of points (zeros of the denominator) from .

## Corollary

Any rational function is analytic.

A particularly simple example of a rational function is , where is a positive integer. This can be differentiated by means of the Reciprocal Rule:

If is used to denote the negative integer , then we can write and . In this form, it is apparent that the formula for differentiating a negative integer power is the same as the formula for differentiating a positive integer power. The only difference is that for negative powers, 0 is excluded from the domain. We state these observations as a final corollary to Theorem 3.

## Corollary

Let . The function has derivative

The domain of is if and if .

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