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.
Answer
Differentiating
with respect to
while keeping
fixed, we obtain
Differentiating with respect to
while keeping
fixed, we obtain
Here
, so
Here
, so
Exercise 20
Calculate the partial derivatives and
of each of the following functions, and evaluate these partial derivatives at
.
Answer
Here
, so
So, at
the partial derivatives have the values
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.
Answer
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.
Answer
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.
Answer
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.
Answer
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,
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,
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