Problem 45 - Calculus for Biology and Medicine

Question

Find the indicated partial derivatives. \(f(x, y)=x \cos y ; \frac{\partial^{2} f}{\partial x d y}\)

Step-by-Step Solution

Answer

\(-\sin y\)

Step 1: Understand the Function and Variables

We start with the function \(f(x, y) = x \cos y\). The function is in terms of two variables, \(x\) and \(y\). The task requires finding the mixed partial derivative \(\frac{\partial^{2} f}{\partial x \partial y}\) which involves differentiating with respect to \(y\) first and then \(x\).

Step 2: Find the First Partial Derivative with Respect to y

To find the first partial derivative \(\frac{\partial f}{\partial y}\), treat \(x\) as a constant and differentiate \(\cos y\) with respect to \(y\). The derivative of \(\cos y\) is \(-\sin y\). Thus, \(\frac{\partial f}{\partial y} = x (-\sin y) = -x \sin y\).

Step 3: Find the Second Partial Derivative with Respect to x

To find the second partial derivative \(\frac{\partial^2 f}{\partial x \partial y}\), differentiate \(-x \sin y\) with respect to \(x\). Treat \(-\sin y\) as a constant while differentiating \(-x \sin y\) with respect to \(x\). The derivative of \(x\) with respect to \(x\) is 1, so \(\frac{\partial^2 f}{\partial x \partial y} = -\sin y\).

Key Concepts

Mixed Partial Derivative

In calculus, mixed partial derivatives provide critical insight into the behavior of functions with multiple variables. A mixed partial derivative, like \(\frac{\partial^2 f}{\partial x \partial y}\), requires us to take the derivative with respect to one variable and then with respect to another. This gives us a measure of how the rate of change of the function with respect to the second variable changes when we vary the first variable.
It is vital to remember the order of differentiation. Here, for the function \(f(x, y) = x \cos y\), the order means we first find the partial derivative with respect to \(y\), treating \(x\) as a constant. Then, we differentiate the result with respect to \(x\).
Mixed partial derivatives elegantly showcase how multivariable functions can change in complex scenarios, providing tools to explore gradients and optimization problems in higher dimensions.

Differentiation

Differentiation is a fundamental tool in calculus used to determine how a function changes. In multivariable calculus, we expand this concept to find partial derivatives. These are derivatives concerning one variable while treating other variables as constants.
When differentiating \(f(x, y) = x \cos y\), we first treat \(x\) as a constant and focus on \(\cos y\). The first partial derivative with respect to \(y\), denoted \(\frac{\partial f}{\partial y}\), for our function is calculated by differentiating \(\cos y\), which results in \(-\sin y\). Consequently, the derivative becomes \(-x \sin y\).
The idea of treating one variable as a constant is crucial. It simplifies the problem and allows mathematicians to focus on one direction of change at a time. Through this process, differentiation becomes a powerful method to analyze changes across multiple dimensions.

Multivariable Calculus

Multivariable calculus extends the basic concepts of single-variable calculus to functions of several variables. It is pivotal in examining how these functions behave and interact in multi-dimensional space.
Functions like \(f(x, y) = x \cos y\) are an example where such calculus is applicable. Unlike single-variable calculus, where we only worry about one direction of change, multivariable calculus requires us to think in multiple directions simultaneously.
By employing techniques like partial and mixed partial derivatives, multivariable calculus provides deeper insight into the multidirectional movement and tendencies of a function. This branch of calculus is essential in fields such as physics, engineering, and economics, where real-world problems often involve multiple varying factors affecting an outcome.
Various applications range from optimizing surfaces and volumes to solving complex systems that arise in practical scenarios. Thus, understanding the principles of multivariable calculus allows for analyzing and solving vastly more complex problems.