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
Sum Rule
is differentiable at
, and
Multiple Rule
is differentiable at
, for
, and
Product Rule
is differentiable at
, and
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.
Sum Rule
Multiple Rule
Answer
Let
. Then
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
By the corollary on differentiating polynomial functions, we have
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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