Problem 7 - University Calculus: Early Transcendentals

Question

Find the first and second derivatives. $$w=3 z^{-2}-\frac{1}{z}$$

Step-by-Step Solution

Answer

First derivative: \( -6z^{-3} + z^{-2} \); Second derivative: \( 18z^{-4} - 2z^{-3} \).

Step 1: Rewrite the function for easier differentiation

We have the function \( w = 3z^{-2} - \frac{1}{z} \). Rewriting \( \frac{1}{z} \) in exponential form gives \( z^{-1} \). Now the function is \( w = 3z^{-2} - z^{-1} \).

Step 2: Differentiate to find the first derivative

Use the power rule \( \frac{d}{dz}z^n = nz^{n-1} \) to differentiate. For \( 3z^{-2} \), the derivative is \( -6z^{-3} \), and for \( -z^{-1} \), the derivative is \( z^{-2} \). The first derivative is \( \frac{dw}{dz} = -6z^{-3} + z^{-2} \).

Step 3: Differentiate again to find the second derivative

Differentiate the first derivative \( \frac{dw}{dz} = -6z^{-3} + z^{-2} \). For \( -6z^{-3} \), the derivative is \( 18z^{-4} \), and for \( z^{-2} \), the derivative is \( -2z^{-3} \). The second derivative is \( \frac{d^2w}{dz^2} = 18z^{-4} - 2z^{-3} \).

Key Concepts

Understanding Derivatives

A derivative is a key concept in calculus, often thought of as a tool that measures how a function changes as its input changes. It provides the rate at which one quantity changes relative to another. When you have a function, say with variable \( z \), its derivative tells you the slope of the tangent to the function at any point on its graph.

• The first derivative of a function is particularly useful because it describes the function's rate of change or its velocity.
• This can indicate where the function is increasing or decreasing.
• For the given function \( w = 3z^{-2} - z^{-1} \), the first derivative \( \frac{dw}{dz} \) reveals the function's behavior with respect to \( z \).

To compute derivatives, we follow a set of differentiation rules, and one commonly used rule is the power rule.

The Power Rule

The power rule is a straightforward way to find derivatives of functions with terms in the form \( z^n \). This rule states:
For any function \( z^n \), \( \frac{d}{dz}z^n = nz^{n-1} \).
It's a fast and effective method that allows us to tackle polynomial functions and power functions with ease.

• In our example, we applied the power rule to each term individually.
• For \( 3z^{-2} \), we multiply \( -2 \) by \( 3 \), giving \( -6z^{-3} \).
• Similarly, for \( -z^{-1} \), multiplying by \( -1 \) results in \( z^{-2} \).

This application of the power rule gives the first derivative: \( \frac{dw}{dz} = -6z^{-3} + z^{-2} \).

Exploring the Second Derivative

The second derivative provides even more insights into a function's behavior. It is the derivative of the first derivative, giving us the rate of change of the rate of change. This can help identify points of inflection and the concavity of the graph.

• If the second derivative is positive, the function is concave up, resembling a cup \( \cup \).
• If it's negative, the function is concave down, resembling a cap \( \cap \).
• In our exercise, we take the first derivative \( -6z^{-3} + z^{-2} \) and differentiate each term again.

Applying the power rule once more:

• The derivative of \( -6z^{-3} \) is \( 18z^{-4} \).
• The derivative of \( z^{-2} \) is \( -2z^{-3} \).

Thus, the second derivative is \( \frac{d^2w}{dz^2} = 18z^{-4} - 2z^{-3} \). This helps further describe the curvature of the graph of the function.