Problem 27 - Calculus Early Transcendentals

Question

\(25-30\) Find the differential of the function. $$ m=p^{5} q^{3} $$

Step-by-Step Solution

Answer

The differential is \(dm = 5p^4q^3 \cdot dp + 3p^5q^2 \cdot dq\).

Step 1: Identify the Function and Variables

The function given is \[ m = p^5 q^3 \]where \(m\) is the function of two variables \(p\) and \(q\). Our goal is to find the expression for the differential \(dm\).

Step 2: Apply the Differential Formula

The differential \(dm\) of \(m\) can be found using the total differential formula for a function of two variables: \[ dm = \frac{\partial m}{\partial p}dp + \frac{\partial m}{\partial q}dq \] This involves finding the partial derivatives of \(m\) with respect to \(p\) and \(q\).

Step 3: Find Partial Derivative with Respect to p

Differentiate \(m = p^5 q^3\) with respect to \(p\):\[ \frac{\partial m}{\partial p} = 5p^{4}q^3 \]Here, \(q^3\) is treated as a constant when differentiating with respect to \(p\).

Step 4: Find Partial Derivative with Respect to q

Now, differentiate \(m = p^5 q^3\) with respect to \(q\):\[ \frac{\partial m}{\partial q} = 3p^5q^{2} \]Here, \(p^5\) is treated as a constant when differentiating with respect to \(q\).

Step 5: Substitute into the Differential Formula

Substitute the partial derivatives obtained in steps 3 and 4 into the differential formula:\[ dm = (5p^4q^3)dp + (3p^5q^2)dq \]This is the expression for the differential \(dm\) of the function \(m = p^5 q^3\).

Key Concepts

Partial Derivative

In differential calculus, partial derivatives are crucial for understanding functions of multiple variables. Unlike a regular derivative, which takes into account a single independent variable, partial derivatives consider one variable at a time while keeping others constant.
This is particularly useful when dealing with functions like the one given:

• For the function \( m = p^5 q^3 \), when taking the partial derivative with respect to \( p \), the term \( q^3 \) is treated as a constant. The result is \( \frac{\partial m}{\partial p} = 5p^4q^3 \).
• Similarly, taking the partial derivative with respect to \( q \) treats \( p^5 \) as a constant, resulting in \( \frac{\partial m}{\partial q} = 3p^5q^2 \).

Understanding these derivatives allows us to determine how the function \( m \) changes in response to small changes in either \( p \) or \( q \) independently. This is a foundational skill in calculus and helps in the broader analysis of functions that depend on multiple variables.

Total Differential

The total differential expresses how a function changes in relation to small changes in all its variables. For a function of two variables, like \( m = p^5 q^3 \), the total differential is a sum of its partial derivatives weighted by the changes in each variable.
The formula for the total differential \( dm \) is:\[ dm = \frac{\partial m}{\partial p}dp + \frac{\partial m}{\partial q}dq \]

• The term \( \frac{\partial m}{\partial p}dp \) measures the change in \( m \) due to a change in \( p \), keeping \( q \) constant.
• The term \( \frac{\partial m}{\partial q}dq \) measures the change in \( m \) due to a change in \( q \), keeping \( p \) constant.

In the given problem, substituting the calculated partial derivatives, we obtain: \[ dm = (5p^4q^3)dp + (3p^5q^2)dq \] This equation captures the combined effect of changes in both variables on the function \( m \). The total differential is a powerful tool when approximating function behavior and is extensively used in fields like physics and engineering.

Function of Two Variables

A function of two variables is an expression that relates to two independent variables, providing a dependent output. In this exercise, the function \( m = p^5 q^3 \) illustrates this concept. Here, \( m \) is dependent on both \( p \) and \( q \).
Understanding functions of two variables involves recognizing:

• How each variable affects the function individually, which leads to the concept of partial derivatives.
• How the function changes altogether with the change in both variables, leading up to the total differential.

Functions of two variables can represent surfaces in three-dimensional space. Each point on the surface corresponds to a unique set of values for \( p \) and \( q \), and results in a specific output value for \( m \). This framework is essential for modeling and solving real-world problems involving multiple inputs. Recognizing that a single output depends on more than one variable is the gateway to multivariable calculus, opening doors to complex analysis and solutions.