Problem 23 - Calculus Early Transcendentals

Question

Find the first partial derivatives of the function. $$w=\sin \alpha \cos \beta$$

Step-by-Step Solution

Answer

The partial derivatives are \(\frac{\partial w}{\partial \alpha} = \cos \alpha \cos \beta\) and \(\frac{\partial w}{\partial \beta} = -\sin \alpha \sin \beta\).

Step 1: Identify Variables for Partial Differentiation

The given function is \(w = \sin \alpha \cos \beta\). Here, \(w\) is a function of the two variables \(\alpha\) and \(\beta\). We need to find the partial derivatives of \(w\) with respect to \(\alpha\) and \(\beta\).

Step 2: Partial Derivative with Respect to \(\alpha\)

To find \(\frac{\partial w}{\partial \alpha}\), differentiate \(w\) with respect to \(\alpha\) while treating \(\beta\) as a constant. Using the product rule: \[\frac{\partial w}{\partial \alpha} = \frac{d}{d\alpha}(\sin \alpha) \cdot \cos \beta = \cos \alpha \cdot \cos \beta.\]

Step 3: Partial Derivative with Respect to \(\beta\)

To find \(\frac{\partial w}{\partial \beta}\), differentiate \(w\) with respect to \(\beta\) while treating \(\alpha\) as a constant. Using the product rule: \[\frac{\partial w}{\partial \beta} = \sin \alpha \cdot \frac{d}{d\beta}(\cos \beta) = -\sin \alpha \cdot \sin \beta.\]

Key Concepts

Partial Differentiation

Partial differentiation is a crucial concept in calculus, especially when dealing with functions of multiple variables. It's much like ordinary differentiation, but here, you're focusing on one variable at a time. To clarify, imagine you have a function, say \( w = \sin \alpha \cos \beta \). Here, \( w \) depends on two variables, \( \alpha \) and \( \beta \). Each of these variables can be considered independently to find how small changes in one affect \( w \), while keeping the other fixed. This results in what are called partial derivatives.

• The notation \( \frac{\partial w}{\partial \alpha} \) represents the partial derivative of \( w \) with respect to \( \alpha \), meaning you treat \( \beta \) as a constant during differentiation.
• Similarly, \( \frac{\partial w}{\partial \beta} \) is the partial derivative concerning \( \beta \), where \( \alpha \) is held constant.

By using partial differentiation, you can explore the behavior of functions with several variables, greatly enhancing your ability to analyze real-world scenarios that depend on more than one varying factor.

Product Rule

The product rule is an essential tool in differentiation, particularly when a function consists of products of two or more functions. Suppose you have a function \( f(x) \cdot g(x) \). To differentiate this with respect to \( x \), the product rule tells you to differentiate each function separately, multiply by the other function, and sum the results. In mathematical terms, it's expressed as:\[ \frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x) .\]
In the context of partial differentiation, such as finding \( \frac{\partial }{\partial \alpha}(\sin \alpha \cos \beta) \), the product rule is applied by treating one variable as constant while differentiating the product.

• Differentiate \( \sin \alpha \), resulting in \( \cos \alpha \), and multiply it by the unchanged \( \cos \beta \).
• However, in this specific setup (since \( \beta \) is constant), only one term affects the result.

This approach highlights the adaptability of the product rule beyond single-variable scenarios, providing a clear path to understanding complex multiple-variable functions.

Trigonometric Functions

Trigonometric functions like \( \sin(x) \) and \( \cos(x) \) are fundamental in both mathematics and physics, often used to describe wave patterns, oscillations, and circular movements. In calculus, these functions present specific differentiation rules.

• The derivative of \( \sin(x) \) with respect to \( x \) is \( \cos(x) \).
• The derivative of \( \cos(x) \) is \(-\sin(x) \).

These rules are critical when differentiating expressions involving trigonometric functions, such as \( \sin \alpha \cos \beta \), with respect to one of their angles. By mastering these derivatives, you can easily tackle problems involving oscillatory or rotational behavior.
For instance, when you differentiate \( \sin \alpha \) in the context of partial differentiation, \( \cos \alpha \) becomes the result, aiding in further calculations. Similarly, differentiating \( \cos \beta \) yields \(-\sin \beta \), showing how swiftly these derivatives can change the form of functions. Understanding these concepts ultimately empowers you to analyze and solve complex trigonometric problems in real-world applications.