Problem 42 - Calculus Early Transcendentals

Question

Find the indicated derivative for the following functions. \(\partial z / \partial p,\) where \(z=x / y, x=p+q,\) and \(y=p-q\)

Step-by-Step Solution

Answer

Answer: $$\frac{\partial z}{\partial p} = \frac{-2q}{(p-q)^2}.$$

Step 1: Rewrite z in terms of p and q

To rewrite \(z\) in terms of \(p\) and \(q\), we first substitute the expressions for \(x\) and \(y\) in terms of \(p\) and \(q\) into the equation for \(z\). We have: $$ z=\frac{x}{y}=\frac{p+q}{p-q}. $$

Step 2: Find the partial derivatives of z with respect to p and q

Now we find the partial derivative of \(z\) with respect to \(p\) and \(q\) using the quotient rule: $$ \frac{\partial z}{\partial p} = \frac{(p-q)\frac{\partial}{\partial p}(p+q) - (p+q)\frac{\partial}{\partial p}(p-q)}{(p-q)^2}. $$ Similarly, $$ \frac{\partial z}{\partial q} = \frac{(p-q)\frac{\partial}{\partial q}(p+q) - (p+q)\frac{\partial}{\partial q}(p-q)}{(p-q)^2}. $$

Step 3: Evaluate the partial derivatives of x and y with respect to p and q

Next, we find the partial derivatives of \(x\) and \(y\) with respect to \(p\) and \(q\). We have: $$ \frac{\partial x}{\partial p} = \frac{\partial}{\partial p}(p+q) = 1 \text{ and } \frac{\partial y}{\partial p} = \frac{\partial}{\partial p}(p-q) = 1, $$ $$ \frac{\partial x}{\partial q} = \frac{\partial }{\partial q}(p+q) = 1 \text{ and } \frac{\partial y}{\partial q} = \frac{\partial}{\partial q}(p-q) = -1. $$

Step 4: Substitute the partial derivatives into the expressions for ∂z/∂p and ∂z/∂q

Now we substitute the partial derivatives from Step 3 into the expressions for \(\partial z/\partial p\) and \(\partial z/\partial q\): $$ \frac{\partial z}{\partial p} = \frac{(p-q)(1) - (p+q)(1)}{(p-q)^2} = \frac{-2q}{(p-q)^2}, $$ $$ \frac{\partial z}{\partial q} = \frac{(p-q)(1) - (p+q)(-1)}{(p-q)^2} = \frac{2p}{(p-q)^2}. $$

Step 5: Provide the final answer

The partial derivative of \(z\) with respect to \(p\) is: $$ \frac{\partial z}{\partial p} = \frac{-2q}{(p-q)^2}. $$

Key Concepts

Quotient Rule

When dealing with the derivative of a fraction, especially in multivariable calculus, the quotient rule is a handy tool. It allows us to differentiate functions that are the ratio of two other functions. Consider a function \( z = \frac{u}{v} \). The quotient rule states:
\[ \frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]
This formula helps ensure we account for both the differences and the interactions between the numerator and the denominator. In our exercise, we applied this rule to find the partial derivatives \( \frac{\partial z}{\partial p} \) and \( \frac{\partial z}{\partial q} \), which involved differentiating the expression \( z = \frac{p+q}{p-q} \). The focus is on systematically distinguishing between each component’s contribution to change.

• Numerator Derivative: Differentiate the top function as if it were alone.
• Denominator Derivative: Differentiate the bottom function separately.
• Combine Carefully: Remember to multiply and subtract correctly to follow the formula.

Understanding the quotient rule is vital because it allows us to dissect how both \( p \) and \( q \) impact the structure of \( z \).

Multivariable Calculus

Multivariable calculus expands the concepts of calculus to functions of several variables. In such contexts, you examine functions like \( z = f(x, y) \) where multiple inputs influence the outcome. Because these functions depend on more than one variable, derivatives become partial derivatives to indicate a focus on one variable’s change at a time.
To solve \( \frac{\partial z}{\partial p} \) and \( \frac{\partial z}{\partial q} \) from the given function \( z = \frac{p+q}{p-q} \), we handled the two variables \( p \) and \( q \).

• Partial Derivatives: Differentiating with respect to one variable while treating others as constants.
• Systematic Approach: It involves evaluating each variable's effect independently.

This approach provides insights into how each variable impacts the function independently, crucial in fields like physics and engineering where multiple variables influence outcomes.

Function of Two Variables

A function of two variables like \( z = \frac{x}{y} \), where both \( x \) and \( y \) depend on another set of variables, can be fascinating to analyze. Here, we see \( z = \frac{p+q}{p-q} \) by substituting expressions for \( x \) and \( y \). The aim is to understand how these dependent variables influence the output \( z \).
Functions of two variables are generally represented as \( f(x, y) \).

• Dependency: Direct impact of the variables on the function.
• Graphical Representation: Often visualized as a 3D surface in space.

In practical scenarios, understanding functions of two variables means understanding how change in one variable, while holding others constant, can affect the overall system. This knowledge is applied in various scientific disciplines to model real-world phenomena more accurately.