Problem 34 - Calculus for Biology and Medicine

Question

In Problems \(31-38\), find \(\partial f / \partial x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. $$ f(x, y, z)=\frac{x y z}{x^{2}+y^{2}+z^{2}} $$

Step-by-Step Solution

Answer

\(\partial f / \partial x = \frac{yz(y^2 + z^2 - x^2)}{(x^2 + y^2 + z^2)^2}\); \(\partial f / \partial y = \frac{xz(x^2 + z^2 - y^2)}{(x^2 + y^2 + z^2)^2}\); \(\partial f / \partial z = \frac{xy(x^2 + y^2 - z^2)}{(x^2 + y^2 + z^2)^2}\).

Step 1: Understand Partial Derivatives

We need to find partial derivatives of the function \(f(x, y, z)=\frac{x y z}{x^2+y^2+z^2}\) with respect to \(x\), \(y\), and \(z\). A partial derivative with respect to a variable means treating all other variables as constants.

Step 2: Apply Quotient Rule for \(\partial f / \partial x\)

To find \(\partial f / \partial x\), use the quotient rule: \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\,du/dx - u\,dv/dx}{v^2}\). Here, \(u = xyz\) and \(v = x^2 + y^2 + z^2\). Evaluate \(\frac{\partial u}{\partial x} = yz\) and \(\frac{\partial v}{\partial x} = 2x\). Substitute into the quotient rule formula.

Step 3: Simplify \(\partial f / \partial x\)

Substitute and simplify: \[\partial f / \partial x = \frac{(x^2 + y^2 + z^2) \cdot yz - xyz \cdot 2x}{(x^2 + y^2 + z^2)^2} = \frac{yz(x^2 + y^2 + z^2 - 2x^2)}{(x^2 + y^2 + z^2)^2}\]. Further simplification gives \[\partial f / \partial x = \frac{yz(y^2 + z^2 - x^2)}{(x^2 + y^2 + z^2)^2}\].

Step 4: Compute \(\partial f / \partial y\) Using Symmetry

Follow the same approach using the quotient rule for \(\partial f / \partial y\). Here \(\frac{\partial u}{\partial y} = xz\) and \(\frac{\partial v}{\partial y} = 2y\). Substitute into the quotient rule: \[\partial f / \partial y = \frac{xz(x^2 + y^2 + z^2 - 2y^2)}{(x^2 + y^2 + z^2)^2}\].

Step 5: Compute \(\partial f / \partial z\) Using Symmetry

Similarly, for \(\partial f / \partial z\), \(\frac{\partial u}{\partial z} = xy\) and \(\frac{\partial v}{\partial z} = 2z\). Use the quotient rule: \[\partial f / \partial z = \frac{xy(x^2 + y^2 + z^2 - 2z^2)}{(x^2 + y^2 + z^2)^2}\].

Key Concepts

Quotient Rule

In calculus, the Quotient Rule is a powerful tool used to differentiate functions that are given as a ratio of two other functions. When dealing with functions like \(f(x, y, z) = \frac{xyz}{x^2 + y^2 + z^2}\), the quotient rule is particularly handy.This rule states that the derivative of a quotient \(\left(\frac{u}{v}\right)\) is given by:

• \[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2}\]

When applying this rule, it is crucial to identify \(u\) and \(v\). For instance, in our exercise, \(u = xyz\) and \(v = x^2 + y^2 + z^2\).To find partial derivatives like \(\partial f / \partial x\), we treat \(y\) and \(z\) as constants and employ the quotient rule utilizing derivatives of both \(u\) and \(v\) with respect to \(x\). This systematic approach allows you to handle complex fractions in multivariable functions efficiently.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus into more than one variable. This branch of calculus is vital for analyzing situations where variables influence one another in complex networks, like in physics, economics, and engineering.In multivariable calculus, functions depend on several variables, and you may need to find the rates at which these variables change. Let's look at a function dependent on three variables: \(f(x, y, z) = \frac{xyz}{x^2 + y^2 + z^2}\). This function is governed by multivariable calculus principles, where each variable could affect the output differently.Understanding how these variables interact within this context is crucial. You can analyze the function from the exercise by considering one variable at a time while keeping the others constant. This method helps isolate the impact of each variable, making it easier to comprehend how changes occur in real-world scenarios.

Partial Derivative with Respect to a Variable

Partial derivatives are fundamental in mathematics, especially concerning functions of multiple variables. These derivatives measure the rate of change of a function concerning one variable while treating others as constants.To compute a partial derivative, such as \(\partial f / \partial x\), we focus solely on how a small change in \(x\) affects the function \(f(x, y, z)\), assuming \(y\) and \(z\) are unchanging. This concept is pivotal since it simplifies understanding how specific variables independently influence a system.Apply the quotient rule, for instance, to find the derivatives like \(\partial f / \partial y\) or \(\partial f / \partial z\). Each variable's partial derivative highlights its role in the equation, allowing us to know exactly how much each contributes to the function's behavior as they vary. Grasping this helps not only in theoretical exercises but also in practical applications where control over process variables is necessary.