Problem 38 - 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. $$ f(x, y, z)=y \tan \left(x^{2}+z\right) $$

Step-by-Step Solution

Answer

The partial derivatives are \(\frac{\partial f}{\partial x} = 2xy \sec^2(x^2 + z)\), \(\frac{\partial f}{\partial y} = \tan(x^2 + z)\), and \(\frac{\partial f}{\partial z} = y \sec^2(x^2 + z)\).

Step 1: Understand the Function

We are given the function \(f(x, y, z) = y \tan(x^2 + z)\) and need to find the partial derivatives \(\frac{\partial f}{\partial x}\), \(\frac{\partial f}{\partial y}\), and \(\frac{\partial f}{\partial z}\). These derivatives represent the rate of change of the function \(f\) with respect to each variable while keeping the others constant.

Step 2: Find \(\frac{\partial f}{\partial x}\)

To find \(\frac{\partial f}{\partial x}\), apply the chain rule. The derivative of \(\tan(u)\) is \(\sec^2(u)\), and the derivative of \(x^2\) is \(2x\):\[\frac{\partial f}{\partial x} = y \cdot \frac{d}{dx}\left[\tan(x^2 + z)\right] = y \cdot \sec^2(x^2 + z) \cdot 2x = 2xy \sec^2(x^2 + z)\]

Step 3: Find \(\frac{\partial f}{\partial y}\)

For \(\frac{\partial f}{\partial y}\), treat \(x\) and \(z\) as constants. The term \(y\tan(x^2 + z)\) can be seen as linear in \(y\):\[\frac{\partial f}{\partial y} = \tan(x^2 + z)\]

Step 4: Find \(\frac{\partial f}{\partial z}\)

To find \(\frac{\partial f}{\partial z}\), we again use the chain rule. The derivative of \(\tan(u)\) is \(\sec^2(u)\) and the derivative of \(z\) is \(1\):\[\frac{\partial f}{\partial z} = y \cdot \frac{d}{dz}\left[\tan(x^2 + z)\right] = y \cdot \sec^2(x^2 + z) \cdot 1 = y \sec^2(x^2 + z)\]

Key Concepts

The Chain Rule

The chain rule is a powerful tool in calculus which allows us to find the derivative of a composite function.
In multivariable calculus, it becomes crucial when dealing with functions that depend on more than one variable.
Let's break it down: the chain rule helps us evaluate how a change in one variable affects a function which, in turn, depends on another variable.
In this problem, the function is given as a combination of variables within a trigonometric function:

• Function: \( f(x, y, z) = y \tan(x^2 + z) \)

When applying the chain rule to find the partial derivative with respect to \(x\), we looked at the outer function \(\tan\) and its inner function \(x^2 + z\).
We took the derivative of \(\tan(u)\), which is \(\sec^2(u)\), and then multiplied it by the derivative of the inner function, \(2x\), to complete the chain rule application.
Thus, the partial derivative with respect to \(x\) is: \[ \frac{\partial f}{\partial x} = 2xy \sec^2(x^2 + z) \] Similarly, the chain rule was applied to find the partial derivative concerning \(z\), resulting in:\[ \frac{\partial f}{\partial z} = y \sec^2(x^2 + z) \] By using the chain rule, we can systematically address functions layered within each other, ensuring we correctly account for each component’s impact.

Rate of Change

Understanding the rate of change is essential in grasping derivatives. In simple terms, a derivative gives us the rate at which a function's output changes with respect to changes in its input. In other words, it tells us how one variable affects another.In multivariable calculus, however, this idea extends further since functions can now depend on multiple variables.
For the given function \( f(x, y, z) = y \tan(x^2 + z) \), finding

• \(\frac{\partial f}{\partial x}\)
• \(\frac{\partial f}{\partial y}\)
• \(\frac{\partial f}{\partial z}\)

is about understanding how changing each of these variables individually affects the function when others are held constant.For example: - \(\frac{\partial f}{\partial x}\) finds how changes in \(x\) influence \(f\), given \(y\) and \(z\) are fixed.- \(\frac{\partial f}{\partial z}\) shows how \(z\) influences \(f\) if \(x\) and \(y\) remain constant.This perspective is particularly helpful in fields like physics and engineering where we often work with models involving several variables. It allows us to predict behavior and changes effectively.

Multivariable Calculus

Multivariable calculus is a branch of mathematics that extends concepts from single-variable calculus to more than one variable.
This branch of mathematics is utilized to evaluate functions with several variables, providing richer results and insights.
Functions like \( f(x, y, z) = y \tan(x^2 + z) \) are classic examples of multivariable functions where more than one variable impacts the output.In multivariable calculus, partial derivatives are crucial as they help us understand how a change in one variable affects the function's output while keeping others constant.
This is essential when analyzing situations in real-world applications, such as:

• Tracking how temperature changes over time and space.
• Assessing how varying economic factors affect market outcomes.

By exploring multivariable calculus concepts, you gain the tools needed to dissect complex models and understand the interplay between different elements within a system.For students, mastering these concepts allows you to solve problems involving derivatives of functions with several variables, enhancing your analytical skills in math and science disciplines.