Problem 44 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The partial derivative of the function w with respect to x is \(\frac{\partial w}{\partial x} = -\sin (x+y)\).

Step 1: Apply Chain Rule

Apply the chain rule to differentiate with respect to x. Chain rule states that \(\frac{dw}{dx}=\frac{dw}{dz}\cdot\frac{dz}{dx}.\)

Step 2: Differentiate w with respect to z

Differentiate w with respect to z. We have \(w = \cos z - \cos x\cos y + \sin x\sin y\), so \(dw/dz = -\sin z.\)

Step 3: Differentiate z with respect to x

Differentiate z with respect to x. We have \(z = x + y\), so \(dz/dx = 1.\)

Step 4: Calculate the derivative of w with respect to x

Calculate the derivative of w with respect to x using the chain rule. By substituting from Steps 2 and 3, we get \(\frac{dw}{dx}=-\sin z \cdot 1.\)

Step 5: Simplify the expression

Using the given relationship \(z = x + y\), we can substitute z in the expression: \[ \frac{\partial w}{\partial x}=-\sin (x+y) \] So, the indicated derivative is \(\frac{\partial w}{\partial x} = -\sin (x+y)\).

Key Concepts

Chain Rule

The chain rule is a critical part of calculus. It helps us find the derivative of a composite function. This means a function that is made by combining two or more functions together.
For example, if you have a function that depends on another variable, which in turn depends on another variable, the chain rule can help you navigate this complexity.

• Essentially, it allows you to differentiate the outer function and multiply it by the derivative of the inner function.
• In our exercise, we apply the chain rule to differentiate the function with respect to multiple variables.

To understand this better, remember the connection: when you have a function, say, \(w = \, f(g(x))\), the derivative \(\frac{dw}{dx}\) is found using \(\frac{dw}{dg}\) and \(\frac{dg}{dx}\).
This multiplication gives us the rate of change of the function depending on how much the inner function changes with respect to \(x\).
This approach helps us navigate through complicated functions systematically.

Partial Derivatives

Partial derivatives extend the concept of differentiation to functions of multiple variables.

• They let us explore how a change in one variable affects the function while keeping other variables constant.
• Think about a multi-input machine: if you change one input slightly, how does the output change?

In our example, the function \(w\) depends on three variables: \(x\), \(y\), and \(z\). We find \(\frac{\partial w}{\partial x}\) to see how \(w\) changes when only \(x\) is varied.

• Unlike standard derivatives, in partial derivatives, you treat variables other than the one you're differentiating with respect to as constants.

This idea is crucial in fields like physics, engineering, and economics. It helps analyze how systems behave under changes in certain conditions without altering others.

Differentiation Techniques

Differentiation techniques involve various methods to find the derivative of functions.

• These techniques include standard rules like the product and quotient rules, and more advanced ones like the chain rule for composite functions.
• Understanding these is key to navigating calculus efficiently.

In our exercise, using different techniques helps simplify the process of finding \(\frac{\partial w}{\partial x}\).

• We primarily used the chain rule, but also standard derivatives to find \(\frac{dw}{dz}\) and \(\frac{dz}{dx}\).
• The steps included recognizing how each part of \(w\) changes with \(z\) and \(x\) respectively.

Each technique serves a purpose and is selected based on the structure of the problem. You choose the method that works best with the function you're dealing with. Thus, mastering differentiation techniques is fundamental for solving calculus problems.