Problem 48 - Calculus for Biology and Medicine

Question

In Problems \(39-48\), find the indicated partial derivatives. $$ f(x, y)=\sin (3 x y) ; \frac{\partial^{3} f}{\partial y^{2} \partial x} $$

Step-by-Step Solution

Answer

The third partial derivative is \(-27x^2 \sin(3xy) - 9x \sin(3xy) + 27x^2y \cos(3xy)\).

Step 1: Identify the Function

We are given the function \( f(x, y) = \sin(3xy) \). Our task is to find the partial derivative \( \frac{\partial^{3} f}{\partial y^{2} \partial x} \).

Step 2: First Partial Derivative with Respect to x

Calculate the first partial derivative of \( f(x, y) \) with respect to \( x \).\[\frac{\partial f}{\partial x} = \cos(3xy) \cdot \frac{d}{dx}(3xy) = 3y \cos(3xy).\]

Step 3: Second Partial Derivative with Respect to y

Now, take the partial derivative of \( \frac{\partial f}{\partial x} = 3y \cos(3xy) \) with respect to \( y \).\[\frac{\partial^{2} f}{\partial y \partial x} = 3 \cos(3xy) + 3y(-\sin(3xy))(3x) = 3 \cos(3xy) - 9xy \sin(3xy).\]

Step 4: Third Partial Derivative with Respect to y Again

Finally, take the derivative of \( \frac{\partial^{2} f}{\partial y \partial x} = 3 \cos(3xy) - 9xy \sin(3xy) \) with respect to \( y \) again.\[\frac{\partial^{3} f}{\partial y^{2} \partial x} = -9x \sin(3xy) \cdot 3x - 9x \sin(3xy) + 9xy \cos(3xy) \cdot 3x = -27x^2 \sin(3xy) - 9x \sin(3xy) + 27x^2y \cos(3xy).\]

Key Concepts

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions of multiple variables, such as functions which depend on both \( x \) and \( y \). One of the intriguing aspects of multivariable calculus is the use of partial derivatives. These allow us to examine how a function changes as one variable changes, while holding others constant. This concept is extremely useful in fields where variables are interdependent.

A partial derivative is denoted by the symbol \( \partial \). For example, if we have a function \( f(x, y) \), the partial derivative of \( f \) with respect to \( x \) is represented as \( \frac{\partial f}{\partial x} \). This expression suggests that when calculating the derivative, we view \( y \) as a constant.

In our exercise, we are asked to compute the mixed third-order partial derivative \( \frac{\partial^3 f}{\partial y^2 \partial x} \). This involves differentiating the function first with respect to \( x \) and then twice with respect to \( y \). Such computations reveal how the function behaves under complex interactions of variables, a core part of multivariable calculus.

Trigonometric Derivatives

Understanding derivatives of trigonometric functions is essential in calculus, and especially so in multivariable situations. Trigonometric functions like \( \sin \) and \( \cos \) frequently appear in these problems. Knowing their derivatives allows us to tackle many kinds of calculus issues, including partial derivatives.

For instance, the derivative of \( \sin(u) \) where \( u \) is a function itself is \( \cos(u) \cdot \frac{du}{dx} \). This formula tells us that when differentiating \( \sin \), one must also consider the derivative of the inner function \( u \).

In the problem at hand, the function \( f(x, y) = \sin(3xy) \) involves finding the derivative of \( \sin \) function combined with a product of two variables. The chain rule is key here, allowing us to systematically differentiate composite structures. Partial derivatives in multivariable calculus leverage these trigonometric derivative rules to compute needed results accurately and efficiently.

Calculus for Biology

Partial derivatives are not just abstract concepts; they hold significant applications in biology and other life sciences. These disciplines often have models with several variables—consider predator and prey populations, enzyme concentrations, or other factors like temperature and pH affecting reaction rates.

Calculating partial derivatives can provide insights into how changes in one biological parameter impact another when the system is complex. For example, if you're analyzing the interaction between two species, a partial derivative can tell you how the growth rate of one species changes with respect to a factor affecting it directly, while other environmental conditions are held constant.

In the context of our given exercise, such computations are akin to observing changes in a biological system where more than one variable dynamically influences the outcome. Whether it's the derivation of a differential equation governing a system's behavior or optimizing a biological process, multivariable calculus offers essential tools for biologists to formulate and analyze real-world biological processes.