Problem 32 - 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. $$ f(x, y, z)=x y z $$

Step-by-Step Solution

Answer

\( \frac{\partial f}{\partial x} = yz, \frac{\partial f}{\partial y} = xz, \frac{\partial f}{\partial z} = xy \).

Step 1: Understand the Function

The given function is a multivariable function \( f(x, y, z) = x y z \). It is a product of three variables: \(x\), \(y\), and \(z\). Our task is to find the partial derivatives of this function with respect to each variable: \(x\), \(y\), and \(z\).

Step 2: Find \( \partial f / \partial x \)

To find the partial derivative of \( f \) with respect to \( x \), treat \( y \) and \( z \) as constants. The derivative of \( x \) with respect to \( x \) is \( 1 \), so the result is:\[ \frac{\partial f}{\partial x} = y z \]

Step 3: Find \( \partial f / \partial y \)

To find the partial derivative of \( f \) with respect to \( y \), treat \( x \) and \( z \) as constants. The derivative of \( y \) with respect to \( y \) is \( 1 \), so the result is:\[ \frac{\partial f}{\partial y} = x z \]

Step 4: Find \( \partial f / \partial z \)

To find the partial derivative of \( f \) with respect to \( z \), treat \( x \) and \( y \) as constants. The derivative of \( z \) with respect to \( z \) is \( 1 \), so the result is:\[ \frac{\partial f}{\partial z} = x y \]

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of calculus that handles functions of more than one variable. Such functions might be, for example, a temperature in a room, which depends on three variables: - The location in the room (defined by coordinates,- Time - And maybe other factors like temperature sources.When we explore multivariable functions, we are often interested in how these functions change when we tweak one of the input variables, keeping the others constant. This is where partial derivatives come into play.

In our given function \( f(x, y, z) = xyz \), the function is dependent on three variables: \( x, y, \) and \( z \). By understanding multivariable calculus, we can determine how changes in any one of these variables affect the function. This is crucial for fields such as physics, economics, and engineering, where systems often depend on multiple factors.

Product Rule

The product rule is a key concept from calculus that helps us differentiate functions that are products of two or more simpler functions. However, in the context of partial derivatives, the product rule is not as frequently needed as in single-variable calculus.

For the function \( f(x, y, z) = xyz \), when we find the partial derivative with respect to one variable, such as \( x \), we treat the other variables \( y \) and \( z \) as constants. Thus, the product rule doesn't explicitly apply in this simple case since each differentiation is straightforward. But if the function were more complex, involving sums and differences of product terms based on multiple variables, utilizing the product rule might become necessary to break down the differentiation process.

• The product rule states: if \( u(x) \) and \( v(x) \) are differentiable functions, then \((uv)' = u'v + uv'.\)
• For multivariable functions, ensure to apply this rule when each component is dependent on more than one variable.

Function of Several Variables

A function of several variables is where a function's value depends on multiple inputs. Understanding how these functions behave is crucial for analyzing scenarios where multiple factors influence an outcome.
The function \( f(x, y, z) = xyz \) is a clear example. Each of the variables \( x \), \( y \), and \( z \) contributes to the outcome of the function.

• If you increase \( x \) while keeping \( y \) and \( z \) fixed, the function's value increases in proportion to \( yz \).
• Similarly, changes in \( y \) or \( z \) will affect the function according to the value of the other two constants.

This interaction shows how sensitive functions can be to each input, thus helping model real-world systems where multiple variables come into play, such as modeling heat distribution, financial markets, or biological systems.