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# 1.6 Further exercises

Here are some further exercises to end this section.

## Exercise 11

Use the definition of derivative to find the derivative of the function

### Answer

The function is defined on the whole of . Let . Then

Since is an arbitrary complex number, is differentiable on the whole of , and the derivative is the function

## Exercise 12

Prove the Quotient Rule for differentiation.

### Answer

Let . Then

Using the Combination Rules for limits of functions, the continuity of , and the fact that , we can take limits to obtain

## Exercise 13

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

### Answer

1. By the Combination Rules,

   The domain of is .

2. By the Combination Rules,

   Since , the domain of is .

3. By the Reciprocal Rule,

   The roots of are . The domain of is therefore .

4. By the Sum Rule and the rule for differentiating integer powers,

   The domain of is .

## Exercise 14

Use Strategy B to show that there are no points of at which the function

is differentiable.

### Answer

Consider an arbitrary complex number , where . Let , . Then , and

Now let , . Then , and

Since the two limits do not agree, it follows that fails to be differentiable at each point of .

## Exercise 15

Describe the approximate geometric effect of the function

on a small disc centred at the point 2.

### Answer

To a close approximation, a small disc centred at 2 is mapped by to small disc centred at . In the process, the disc is scaled by the factor and rotated through the angle .

By the Quotient Rule,

So scales the disc by the factor 4 and rotates it anticlockwise through the angle .

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