Problem 33 - Calculus with Concepts in Calculus

Question

Find \(f_{x y}\) and \(f_{y x}\) $$ f(x, y, z)=z \cos x y $$

Step-by-Step Solution

Answer

\(f_{xy} = f_{yx} = -z(x\cos(xy) + \sin(xy))\).

Step 1: Differentiate with Respect to x

To find \(f_{xy}\), we first need to differentiate \(f(x, y, z) = z \cos(xy)\) with respect to \(x\), treating \(y\) and \(z\) as constants. The derivative of \(\cos(xy)\) with respect to \(x\) is \(-y \sin(xy)\). Thus, the partial derivative with respect to \(x\) is \[\frac{\partial f}{\partial x} = z(-y \sin(xy)) = -zy \sin(xy).\]

Step 2: Differentiate Again for f_x_y

Now, differentiate \(-zy \sin(xy)\) with respect to \(y\). Apply the product rule: \[\frac{\partial}{\partial y}(-zy \sin(xy)) = -z \left(\sin(xy) + xy\cos(xy)\right).\]So, \(f_{xy} = -z(x\cos(xy) + \sin(xy))\).

Step 3: Differentiate with Respect to y

Next, we find \(f_{yx}\) by first differentiating \(f(x, y, z) = z \cos(xy)\) with respect to \(y\), treating \(x\) and \(z\) as constants. The derivative of \(\cos(xy)\) with respect to \(y\) is \(-x \sin(xy)\), leading to\[\frac{\partial f}{\partial y} = z(-x \sin(xy)) = -zx \sin(xy).\]

Step 4: Differentiate Again for f_y_x

Differentiate \(-zx \sin(xy)\) with respect to \(x\). Here, apply the chain rule:\[\frac{\partial}{\partial x}(-zx \sin(xy)) = -z \left(\sin(xy) + xy \cos(xy)\right).\]Therefore, \(f_{yx} = -z(x\cos(xy) + \sin(xy))\).

Key Concepts

Partial Differentiation

Partial differentiation is a technique used in calculus to find the rate of change of a multivariable function concerning one specific variable. When performing partial differentiation, we treat all other variables as constants. This is an important concept when dealing with functions that depend on multiple variables, as it allows us to understand how a change in one variable affects the function while all others remain constant.
For example, consider the function \( f(x, y, z) = z \cos(xy) \). If we're tasked with finding the partial derivative of \( f \) with respect to \( x \), we focus solely on the changes caused by \( x \), assuming \( y \) and \( z \) are constants. This process helps simplify the evaluation by reducing the interaction among variables, making the differentiation process more manageable in multivariable calculus.

Chain Rule

The chain rule is a fundamental principle in calculus that helps us differentiate composite functions. It's crucial when dealing with partial differentiation where functions inside other functions are present, especially when variables multiply each other.

• In the provided exercise, the chain rule helps us find derivatives when expressions like \( \,xy\, \) appear, which are inside a trigonometric function.
• If \( \cos(xy) \) is part of \( f(x, y, z) = z \cos(xy) \), then while differentiating, we need to consider how changes in the product \( xy \) affect the result.

For instance, in Step 4 of the solution, when we differentiate \(-zx \sin(xy)\) with respect to \( x \), the chain rule is utilized to acknowledge the contribution of \( xy \) and its influence on \( x\). This rule allows us to systematically break down complex expressions and calculate the required derivative.

Product Rule

The product rule is an essential calculus rule that allows us to differentiate functions that are multiplied together. Specifically, when two or more functions are multiplied, the product rule provides a method to differentiate the resulting expression.

• The rule states: if you have two functions, \( u(x) \) and \( v(x) \), then the derivative of their product, \( u(x)v(x)\), is \( u'(x)v(x) + u(x)v'(x)\).
• Simplified, this means you take the derivative of the first function and multiply it by the second function, then add the product of the first function and the derivative of the second function.

In Step 2 of our solution, when differentiating \(-zy \sin(xy)\) with respect to \( y \), the product rule helps manage expressions where \( y \) appears in both parts of the product. This involves handling both \(-zy\) as a factor and the sine function, allowing us to correctly account for all components when finding \( f_{xy} \). Using this rule effectively is crucial for accurately obtaining mixed partial derivatives.