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 z long right arrow from bar one and z long right arrow from bar z 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 cap a and cap b , respectively, and let alpha be a limit point of cap a intersection cap b . If and are differentiable at alpha , 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 cap a and cap b in Theorem 3 are regions, then every point of cap a intersection cap b is a limit point of cap a and of cap b .

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 alpha . If f of alpha not equals zero , then one solidus f is differentiable at alpha , and

left parenthesis one divided by f right parenthesis super prime times left parenthesis alpha right parenthesis equals negative f super prime of alpha divided by left parenthesis f of alpha right parenthesis squared full stop

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 alpha , then it is continuous at alpha , so lim over z right arrow alpha of g of z equals g of alpha .

Example 2

Prove the Product Rule for differentiation.

Solution

Let cap f equals f times g . Then

multiline equation row 1 Blank lim over z right arrow alpha of cap f of z minus cap f of alpha divided by z minus alpha Blank row 2 Blank equals lim over z right arrow alpha of f of z times g of z minus f of alpha times g of alpha divided by z minus alpha Blank row 3 Blank equals lim over z right arrow alpha of left parenthesis f of z minus f of alpha right parenthesis times g of z plus f of alpha times left parenthesis g of z minus g of alpha right parenthesis divided by z minus alpha Blank row 4 Blank equals left parenthesis lim over z right arrow alpha of f of z minus f of alpha divided by z minus alpha right parenthesis times left parenthesis lim over z right arrow alpha of g of z right parenthesis plus f of alpha times left parenthesis lim over z right arrow alpha of g of z minus g of alpha divided by z minus alpha right parenthesis Blank row 5 Blank equals f super prime of alpha times g of alpha plus f of alpha times g super prime of alpha full stop Blank

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 f of z equals z is entire with derivative f super prime of z equals one , we can use the Product Rule repeatedly to show that the function

f of z equals z super n times left parenthesis z element of double-struck cap c right parenthesis

is entire, and that its derivative is

f super prime of z equals n times z super n minus one times left parenthesis z element of double-struck cap c right parenthesis full stop

(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,

if f of z equals sum with 3 summands z super four minus three times z squared plus two times z plus one comma then f super prime of z equals four times z cubed minus six times z plus two full stop

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

Corollary Differentiating Polynomial Functions

Let be the polynomial function

p of z equals sum with variable number of summands a sub n times z super n plus ellipsis plus a sub two times z squared plus a sub one times z plus a sub zero times left parenthesis z element of double-struck cap c right parenthesis comma

where a sub zero comma a sub one comma ellipsis comma a sub n element of double-struck cap c and a sub n not equals zero . Then is entire with derivative

p super prime of z equals sum with variable number of summands n times a sub n times z super n minus one plus ellipsis plus two times a sub two times z plus a sub one times left parenthesis z element of double-struck cap c right parenthesis full stop

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

f of z equals two times z squared plus z divided by z squared plus one comma

and specify its domain.

Solution

By the corollary on differentiating polynomial functions, the derivative of z long right arrow from bar two times z squared plus z is

z long right arrow from bar four times z plus one comma

and the derivative of z long right arrow from bar z squared plus one is

z long right arrow from bar two times z full stop

Provided that z squared plus one is non-zero, we can apply the Quotient Rule to obtain

equation sequence part 1 f super prime of z equals part 2 left parenthesis z squared plus one right parenthesis times left parenthesis four times z plus one right parenthesis minus left parenthesis two times z squared plus z right parenthesis times left parenthesis two times z right parenthesis divided by left parenthesis z squared plus one right parenthesis squared equals part 3 sum with 3 summands negative z squared plus four times z plus one divided by left parenthesis z squared plus one right parenthesis squared full stop

Since z squared plus one is non-zero everywhere apart from and negative i , it follows that the domain of f super prime is double-struck cap c minus i comma negative i .

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 double-struck cap c .

Corollary

Any rational function is analytic.

A particularly simple example of a rational function is f of z equals one solidus z super n , where is a positive integer. This can be differentiated by means of the Reciprocal Rule:

equation sequence part 1 f super prime of z equals part 2 negative n times z super n minus one divided by left parenthesis z super n right parenthesis squared equals part 3 negative n times z super negative n minus one full stop

If is used to denote the negative integer negative n , then we can write f of z equals z super k and f super prime of z equals k times z super k minus one . 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 k element of double-struck cap z minus zero . The function f of z equals z super k has derivative

f super prime of z equals k times z super k minus one full stop

The domain of f super prime is double-struck cap c if k greater than zero and double-struck cap c minus zero if k less than zero .

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