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 .
a.
b.
c.
d.
Answer
a.By the Combination Rules,
The domain of is .
b.By the Combination Rules,
Since , the domain of is .
c.By the Reciprocal Rule,
The roots of are . The domain of is therefore .
d.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 .