Problem 31 - Calculus an Applied Approach

Question

Find the first partial derivatives with respect to \(x, y,\) and \(z .\) $$ w=\frac{2 z}{x+y} $$

Step-by-Step Solution

Answer

The first partial derivatives of \(w\) with respect to \(x, y, z\) are \(-\frac{2z}{(x+y)^2}, -\frac{2z}{(x+y)^2}\), and \(\frac{2}{x+y}\) respectively.

Step 1: Identify the North Function

In this problem, the function is \(w=\frac{2 z}{x+y}\).

Step 2: Find the Partial Derivative with respect to \(x\)

To find the partial derivative of \(w\) with respect to \(x\), treat \(y\) and \(z\) as constants and apply the product rule and quotient rule of differentiation. This yields: \(-\frac{2z}{(x+y)^2}\)

Step 3: Find the Partial Derivative with respect to \(y\)

To find the partial derivative of \(w\) with respect to \(y\), treat \(x\) and \(z\) as constants and apply the product rule and quotient rule of differentiation. This yields: \(-\frac{2z}{(x+y)^2}\).

Step 4: Find the Partial Derivative with respect to \(z\)

To find the partial derivative of \(w\) with respect to \(z\), treat \(x\) and \(y\) as constants and apply the product rule and quotient rule of differentiation. This yields: \(\frac{2}{x+y}\).

Key Concepts

Multivariable calculus

Multivariable calculus extends the principles of calculus to functions of more than one variable. This means working with functions that have two, three, or even more variables. These are known as multivariable functions, such as the example function given, \[ w = \frac{2z}{x+y} \].

When dealing with multivariable functions, partial derivatives play a crucial role. Unlike standard derivatives, partial derivatives are taken with respect to one variable while treating all other variables as constants.

• Partial derivative with respect to \(x\): Here, \(y\) and \(z\) are considered constants.
• Partial derivative with respect to \(y\): Here, \(x\) and \(z\) are considered constants.
• Partial derivative with respect to \(z\): Here, \(x\) and \(y\) are considered constants.

Each partial derivative measures how the function \(w\) changes as we vary a single variable while keeping others fixed, offering insights into the behavior of multivariable functions.

Quotient rule

The quotient rule is a method used in calculus for finding the derivative of a quotient of two functions, which comes in handy when differentiating expressions in multivariable functions like \[ w = \frac{2z}{x+y} \].

Essentially, if you have a function \( \frac{u}{v} \), where both \(u\) and \(v\) are functions of a variable, the quotient rule states:\[\left(\frac{u}{v}\right)' = \frac{v \cdot u' - u \cdot v'}{v^2}.\]

• Here, \(u = 2z\) and \(v = x+y\).
• When applying the rule, differentiate \(u\) and \(v\) separately first.
• Then apply the formula correctly to get the required partial derivative.

By applying this rule to our partial derivative problems, it helps efficiently determine how the function behaves as one of the independent variables changes. Whether it’s with respect to \(x\), \(y\), or \(z\), the quotient rule simplifies the process by handling these fractional forms.

Product rule

The product rule in calculus is utilized for finding the derivative of the product of two functions. While it might not always be explicitly mentioned, it often pairs with the quotient rule when both concepts are simultaneously needed.

The product rule states:\[(uv)' = u'v + uv',\]where \(u\) and \(v\) are functions of a variable.

In the context of multivariable calculus for the function \[ w = \frac{2z}{x+y} \], even though it's primarily about division, recognizing that multiplication and division are interlinked helps when taking derivatives.

• This rule ensures each part of the product is considered differentiatively.
• Such differentiation assures accuracy, especially in more complex multivariable functions.
• The product rule complements the mechanics of the quotient rule, making overall differentiation more seamless.

By keeping these rules in mind, you can better tackle any derivative problem, enhancing your understanding and making the task more manageable.