Problem 26 - Advanced Engineering Mathematics

Question

$$f^{\prime}(z)=\lim _{\Delta z \rightarrow 0} \frac{\frac{1}{z+\Delta z}-\frac{1}{z}}{\Delta z}=$$ $$\lim _{\Delta z \rightarrow 0} \frac{-\Delta z}{(\Delta z) z(z+\Delta z)}=\lim _{\Delta z \rightarrow 0} \frac{-1}{z(z+\Delta z)}=-\frac{1}{z^{2}}$$

Step-by-Step Solution

Answer

The derivative of \( f(z) = \frac{1}{z} \) is \( f'(z) = -\frac{1}{z^2} \).

Step 1: Conceptualize the Problem

We need to find the derivative of the function \( f(z) = \frac{1}{z} \) with respect to \( z \), using the definition of the derivative as the limit of the difference quotient.

Step 2: Set up the Difference Quotient

The difference quotient for the derivative \( f'(z) \) is given by \( \frac{f(z+\Delta z) - f(z)}{\Delta z} \). For \( f(z) = \frac{1}{z} \), substitute to get: \[ \frac{\frac{1}{z+\Delta z} - \frac{1}{z}}{\Delta z} \].

Step 3: Simplify the Fraction

To simplify \( \frac{1}{z+\Delta z} - \frac{1}{z} \), combine into a single fraction: \[ \frac{z - (z+\Delta z)}{z(z+\Delta z)} = \frac{-\Delta z}{z(z+\Delta z)} \].

Step 4: Evaluate the Limit

Substitute the simplified fraction back into the difference quotient: \( \frac{-\Delta z}{\Delta z \cdot z(z+\Delta z)} \), which reduces to \( \frac{-1}{z(z+\Delta z)} \). Evaluate the limit as \( \Delta z \to 0 \): \[ \lim_{\Delta z \to 0} \frac{-1}{z(z + \Delta z)} = \frac{-1}{z^2} \].

Key Concepts

Derivative

The term "derivative" in calculus represents the rate at which a function is changing at any given point. It's like finding the slope of a function or curve at a specific point. In mathematical terms, if you have a function, say \( f(x) \), the derivative, denoted \( f'(x) \) or \( \frac{df}{dx} \), helps us understand how \( f(x) \) reacts to tiny changes in \( x \).
The derivative is particularly useful in understanding the behavior of graphs and in predicting how trends will develop.

• An intuitive way to think about derivatives is as the velocity of a car on a road – it tells you how fast your speed or position is changing.
• In more everyday life situations, it's like finding out how fast a movie ticket price is increasing over time.

Difference Quotient

The difference quotient is a key concept in finding the derivative of a function. It is essentially a formula that tells us the average rate of change of the function over a small interval. It is expressed as \( \frac{f(x + \Delta x) - f(x)}{\Delta x} \).
This formula helps us approximate the derivative by calculating how much \( f(x) \) changes when \( x \) is increased by a small amount \( \Delta x \).

• We use \( \Delta x \) to represent a small change in \( x \), and similarly, \( \Delta z \) for complex variables.
• As we make \( \Delta x \) smaller and smaller, the difference quotient approaches the actual derivative.

Limit

In calculus, the concept of a limit is crucial to defining derivatives, and other important calculus concepts. The limit of a function describes the value that a function approaches as the input, or variable, approaches a specific value.
For instance, when we say \( \lim_{x \to a} f(x) = L \), it means as \( x \) gets closer to \( a \), \( f(x) \) approaches \( L \).

• Limits are key when finding the derivative as they allow us to consider what happens to the function as the change \( \Delta x \) becomes infinitesimally small (approaches zero).
• Without limits, we wouldn't be able to perform many of the computations that calculus allows us to do.

Function of a Complex Variable

A function of a complex variable is slightly different from regular functions because they use complex numbers as inputs. When dealing with functions like \( f(z) \) where \( z \) represents a complex number (\( z = x + yi \)), calculus still applies but with unique properties that cater to complex numbers.
For instance, if \( z \) is a complex variable, we might look at \( f(z) = \frac{1}{z} \) and explore its derivative using complex calculus tools

• Complex variable functions are vital in fields like engineering and physics, where real-world phenomena can often be modeled more accurately using them.
• The processes and rules for differentiating these functions follow ideas similar to real functions but integrate the elegance of complex numbers to analyze them.