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# 2.3 Further exercises

Here are some further exercises to end this section.

## Exercise 19

Calculate the partial derivatives and of each of the following functions.

• a.

• b.

• c.

• a.Differentiating with respect to while keeping fixed, we obtain

Differentiating with respect to while keeping fixed, we obtain

• b.Here , so

• c.Here , so

## Exercise 20

Calculate the partial derivatives and of each of the following functions, and evaluate these partial derivatives at .

• a.

• b.

• a.Here , so

So, at the partial derivatives have the values

• b.Here , so

So, at the partial derivatives have the values

## Exercise 21

Find the gradient of the graph of at the point in the -direction and in the -direction.

Since , it follows that

The gradient of the graph at in the -direction is

The gradient of the graph at in the -direction is

## Exercise 22

Use the Cauchy–Riemann equations to show that there is no point of at which the function

is differentiable.

Writing in the form

we obtain

Hence

If is differentiable at , then the Cauchy–Riemann equations require that

that is,

But is never zero, so , which is impossible. It follows that there is no point of at which is differentiable.

## Exercise 23

Use the Cauchy–Riemann equations to show that the function

is entire, and find its derivative.

In this case,

so

These partial derivatives are defined and continuous on the whole of . Furthermore,

so the Cauchy–Riemann equations are satisfied at every point of .

By the Cauchy–Riemann Converse Theorem, is entire, and

(So , and in fact .)

## Exercise 24

Use the Cauchy–Riemann equations to find all the points at which the following functions are differentiable, and calculate their derivatives.

• a.

• b.

• a.Here

so

The Cauchy–Riemann equations are satisfied only if . So cannot be differentiable at unless . Since the partial derivatives above exist, and are continuous on (and in particular when ), it follows from the Cauchy–Riemann Converse Theorem that  is differentiable on the set .

On this set,

• b.Here

so

The Cauchy–Riemann equations are not satisfied unless and . So is not differentiable except possibly at 0. Since the partial derivatives above exist, and are continuous at , it follows from the Cauchy–Riemann Converse Theorem that  is differentiable at 0. Furthermore,