Problem 118 - Calculus Volume 3

Question

For the following exercises, calculate the partial derivatives. \(\quad \frac{\partial z}{\partial x}\) for \(z=\sin (3 x) \cos (3 y)\).

Step-by-Step Solution

Answer

\( \frac{\partial z}{\partial x} = 3\cos(3x)\cos(3y) \)

Step 1: Identify the Function Components

The given function is \( z = \sin(3x)\cos(3y) \). This is a product of two functions, \( u = \sin(3x) \) and \( v = \cos(3y) \). We will need to apply the product rule for differentiation.

Step 2: Apply the Product Rule

Recall the product rule for differentiation: if \( z = u \cdot v \), then \( \frac{\partial z}{\partial x} = \frac{\partial u}{\partial x} \cdot v + u \cdot \frac{\partial v}{\partial x} \). Let's calculate each part separately.

Step 3: Differentiate \( u = \sin(3x) \) with Respect to \( x \)

To find \( \frac{\partial u}{\partial x} \), use the chain rule: The derivative of \( \sin(3x) \) with respect to \( x \) is \( 3\cos(3x) \). Thus, \( \frac{\partial u}{\partial x} = 3\cos(3x) \).

Step 4: Differentiate \( v = \cos(3y) \) with Respect to \( x \)

The function \( v = \cos(3y) \) depends only on \( y \); therefore, its derivative with respect to \( x \) is 0. So, \( \frac{\partial v}{\partial x} = 0 \).

Step 5: Combine Results Using the Product Rule

Using the product rule, substitute the results into the formula: \( \frac{\partial z}{\partial x} = (3\cos(3x)) \cdot \cos(3y) + \sin(3x) \cdot 0 \). This simplifies to \( \frac{\partial z}{\partial x} = 3\cos(3x)\cos(3y) \).

Key Concepts

Understanding the Chain Rule

When we talk about the chain rule in calculus, we refer to a fundamental method used to differentiate composite functions. A composite function is essentially a function within another function.
This occurs frequently when we deal with how changes in one variable affect another, especially in terms of inner and outer functions in mathematics. Let’s simplify this with an example.

• Imagine the function \(f(g(x))\), where \(g(x)\) lies within \(f(x)\).
• To compute the derivative, you first differentiate the outer function \(f\) with respect to \(g(x)\).
• Then, multiply that by the derivative of the inside function, \(g(x)\).

In our original exercise, we encounter a similar situation with \(\sin(3x)\). Here, \(3x\)\ is the inside function within \(\sin(x)\),\ and by applying the chain rule, we find the derivative to be \(3\cos(3x)\).
Mastering the chain rule is essential as it is pervasive in both simple and complex calculus problems, serving as a tool to unravel multi-layered functional relationships.

Decoding the Product Rule

The product rule is a core concept when dealing with the differentiation of products of two or more functions. When we refer to functions in a product form, we're talking about expressions like \(u \cdot v\).
Here’s how to approach it:

• The formula derived from the product rule is \(\frac{\partial}{\partial x} \, (u \v) = \frac{\partial u}{\partial x} \cdot v + u \cdot \frac{\partial v}{\partial x}\).
• This tells us to differentiate each function with respect to the variable, one by one, while holding the others constant, then multiply and combine them.

In the given exercise, identify \(u\) as \(\sin(3x)\) and \(v\) as \(\cos(3y)\). Notice, this expression is a multiplication of \(u\) and \(v\), revealing the necessity of the product rule.
Apply the rule: \(\frac{\partial z}{\partial x} = \left(\frac{\partial u}{\partial x}\right) \cdot v + u \cdot \frac{\partial v}{\partial x} \= (3\cos(3x)) \cdot \cos(3y) + \sin(3x) \cdot 0\).
Thus, the derivative simplifies to \(3\cos(3x)\cos(3y)\).
Essentially, the product rule is crucial for understanding how to handle derivatives of multiplied functions efficiently.

Executing Differentiation Steps

Differentiation is a process that leads to the derivative, a concept pivotal in understanding change. The steps of differentiation involve breaking down a function to derive a new function that describes its rate of change.
Let's consider the essential steps to differentiate the function given in our exercise.

• **Step 1:** Identify the form of the function, \(z = \sin(3x)\cos(3y)\). Recognize that this involves both the product and chain rules.
• **Step 2:** Apply the product rule, which requires differentiating one function while keeping the other constant.
• **Step 3:** Use the chain rule where necessary, such as when differentiating \(\sin(3x)\) to obtain \(3\cos(3x)\).
• **Step 4:** Simplify your expression. In our equation, \(\frac{\partial v}{\partial x}\) is zero
• **Final Step:** Combine results as per the product rule.

These steps reveal the derivatives provide insight into the function's behavior by describing how variables influence these changes under certain conditions.
Efficiently executing these steps ensures accurate results, facilitating deeper mathematical insight.