Problem 31 - Calculus for Biology and Medicine

Question

Find \(\partial f / 2 x, \partial f / \partial y\), and \(\partial f / \partial z\) for the given functions. \(f(x, y, z)=x^{2} z+y z^{2}-x y\)

Step-by-Step Solution

Answer

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

Step 1: Identify Partial Derivative with Respect to x

The first task is to find the partial derivative of the function \(f(x, y, z) = x^{2}z + yz^{2} - xy\) with respect to \(x\). Consider \(y\) and \(z\) as constants while differentiating with respect to \(x\). The derivative of \(x^2z\) with respect to \(x\) is \(2xz\), the derivative of \(yz^2\) with respect to \(x\) is 0 (since it's a constant term in this context), and the derivative of \(-xy\) with respect to \(x\) is \(-y\). So the partial derivative \(\frac{\partial f}{\partial x} = 2xz - y.\)

Step 2: Identify Partial Derivative with Respect to y

Now find the partial derivative of the function \(f(x, y, z) = x^{2}z + yz^{2} - xy\) with respect to \(y\). Consider \(x\) and \(z\) as constants while differentiating with respect to \(y\). The derivative of \(x^2z\) with respect to \(y\) is 0, the derivative of \(yz^2\) with respect to \(y\) is \(z^2\), and the derivative of \(-xy\) with respect to \(y\) is \(-x\). Therefore, the partial derivative \(\frac{\partial f}{\partial y} = z^2 - x.\)

Step 3: Identify Partial Derivative with Respect to z

Finally, find the partial derivative of the function \(f(x, y, z) = x^{2}z + yz^{2} - xy\) with respect to \(z\). Consider \(x\) and \(y\) as constants while differentiating with respect to \(z\). The derivative of \(x^2z\) with respect to \(z\) is \(x^2\), the derivative of \(yz^2\) with respect to \(z\) is \(2yz\), and the derivative of \(-xy\) with respect to \(z\) is 0. Thus, the partial derivative \(\frac{\partial f}{\partial z} = x^2 + 2yz.\)

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of traditional calculus that takes into account functions of more than one variable. Unlike single-variable calculus, which deals with functions of a single variable, multivariable calculus focuses on functions of two or more independent variables. In the given exercise, the function is defined in terms of three variables: \(x\), \(y\), and \(z\).
The goal of multivariable calculus is to understand how these different variables interact and affect each other. It extends the concept of derivative and integral to higher dimensions. This involves working with partial derivatives, gradients, and multiple integrals.

• Partial differentiation is often used to find rates of change for systems under the influence of several factors.
• Understanding how to differentiate functions of several variables is crucial in fields like engineering and physics, where processes depend on more than one factor.

With multivariable calculus, we aim to analyze and visualize scenarios where more than one element plays a pivotal role in shaping outcomes.

Partial Differentiation

Partial differentiation is a technique used in multivariable calculus to find the derivative of a function concerning one variable while keeping the other variables constant. This process helps determine how a function changes as one specific variable changes.
In the exercise, we calculated the partial derivatives of the function \(f(x, y, z) = x^2z + yz^2 - xy\) with respect to \(x\), \(y\), and \(z\).
Here's a recap:

• When finding the partial derivative with respect to \(x\), consider \(y\) and \(z\) as constants. The result was \(\frac{\partial f}{\partial x} = 2xz - y\).
• For the partial derivative with respect to \(y\), treat \(x\) and \(z\) as constants. We found \(\frac{\partial f}{\partial y} = z^2 - x\).
• The partial derivative with respect to \(z\) is found by holding \(x\) and \(y\) constant. This yielded \(\frac{\partial f}{\partial z} = x^2 + 2yz\).

These derivatives tell us how the function changes when only one of the variables is varied while others stay constant. They are widely used in optimization problems and modeling phenomena in the natural sciences.

Functions of Several Variables

Functions of several variables are expressions that depend on more than one input. In mathematics, these functions are essential because they model situations where multiple factors impact results. For instance, in a physical system, a function might relate variables like temperature, pressure, and volume.
The given function in the exercise is \(f(x, y, z) = x^2z + yz^2 - xy\), where \(x\), \(y\), and \(z\) are independent variables. In practical terms, consider how each variable affects the function both individually and collectively.
When examining such functions:

• Visualize them using surfaces or contours to grasp how variables interact over a region.
• Predict behavior by approximating nearby values using derivatives, which addresses the rate of change.
• Solve real-world problems by understanding how each variable impacts the whole system when tweaking one or more elements.

Understanding and analyzing functions with multiple variables allows us to gain insight into complex systems where several factors are intertwined.