Problem 41 - Calculus for Biology and Medicine

Question

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

Step-by-Step Solution

Answer

The mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \) is \( e^y \).

Step 1: Identify the function and order of derivatives

The function provided is \( f(x, y) = x e^y \). We are required to find the mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \), which means we first differentiate with respect to \( y \) and then with respect to \( x \).

Step 2: First Partial Derivative with Respect to y

To find \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant. Differentiate \( f(x, y) = x e^y \) with respect to \( y \). The derivative of \( e^y \) with respect to \( y \) is \( e^y \), so we get:\[ \frac{\partial f}{\partial y} = x e^y \]

Step 3: Second Partial Derivative with Respect to x

Now, take the derivative of \( \frac{\partial f}{\partial y} = x e^y \) with respect to \( x \). Here, \( e^y \) is treated as a constant because it's independent of \( x \). Hence, the derivative is:\[ \frac{\partial^2 f}{\partial x \partial y} = e^y \]

Step 4: Simplify and Conclude

The mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \) is found to be \( e^y \). Since no further simplification is needed, this completes the calculation.

Key Concepts

Mixed Derivatives

Mixed derivatives involve taking partial derivatives of a multivariable function with respect to two or more different variables in succession. In many cases, these derivatives are crucial in understanding the behavior of functions in multiple dimensions. Let's consider the function from the exercise, \( f(x, y) = x e^y \). The mixed partial derivative \( \frac{\partial^2 f}{\partial x \partial y} \) requires two steps:

• First, take the derivative with respect to \( y \), treating \( x \) as a constant.
• Second, derive the result with respect to \( x \), treating \( y \) as a constant this time.

This process allows us to explore how the function changes as each variable changes. The order in which derivatives are taken can affect the outcome unless the function is particularly nice (continuous and its derivatives follow certain conditions), as per Clairaut's theorem. However, for most functions encountered in calculus, mixed partial derivatives in different orders are equal.

Differentiation with Respect to Variables

Differentiation with respect to variables refers to the process of computing a derivative by holding all other variables constant. This is a foundational concept in calculus, especially in multivariable calculus where functions involve more than one variable.

When working with a function like \( f(x, y) = x e^y \), we focus on one variable at a time. For example:

• When differentiating with respect to \( y \), think of \( x \) as though it were a number, unchanging as you derive \( e^y \) to get \( x e^y \).
• Next, when differentiating with respect to \( x \), treat \( e^y \) as a constant because it doesn’t depend on \( x \), yielding a derivative of \( e^y \).

This approach simplifies dealing with changes in each variable independently, making it manageable to tackle complex multivariable problems, and is vital for solving real-world problems where multiple factors influence outcomes.

Calculus in Biology

Calculus is not only an essential mathematical tool but also pivotal in biological sciences. In biology, differential calculus is used to study changes and dynamics in biological systems. Partial derivatives, like the mixed derivatives we've discussed, help model how changes in one biological factor impact another over time or across conditions.

• In population biology, partial derivatives can track how population size changes with varying environmental factors like food supply and habitat space.
• In pharmacokinetics, they're used to observe how drugs disperse in different tissues, depending on dosage and frequency.

Understanding these relationships is crucial for predicting biological outcomes and designing interventions. The precise calculations allow biologists to anticipate changes and make informed decisions about research and treatment plans. This intersection of calculus and biology underscores the importance of mathematical literacy in solving complex biological questions.