Problem 3 - Calculus for Biology and Medicine

Question

In Problems 1-16, find \(\partial f / \partial x\) and \(\partial f / \partial y\) for the given functions. $$ f(x, y)=(x y)^{3 / 2}-(x y)^{2 / 3} $$

Step-by-Step Solution

Answer

The partial derivatives are \( \frac{\partial f}{\partial x} = \frac{3}{2} y (xy)^{1/2} - \frac{2}{3} y (xy)^{-1/3} \) and \( \frac{\partial f}{\partial y} = \frac{3}{2} x (xy)^{1/2} - \frac{2}{3} x (xy)^{-1/3} \).

Step 1: Define and Differentiate

We are given the function \( f(x, y) = (xy)^{3/2} - (xy)^{2/3} \). We need to find the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \). Partial derivatives require us to differentiate with respect to one variable while keeping the other constant.

Step 2: Differentiate with Respect to \( x \)

First, we find \( \frac{\partial f}{\partial x} \):- The first term is \( (xy)^{3/2} \). Using the chain rule, we differentiate: \( \frac{d}{dx} (xy)^{3/2} = \frac{3}{2} (xy)^{1/2} \cdot y \).- The second term is \( (xy)^{2/3} \). Again using the chain rule, we have: \( \frac{d}{dx} (xy)^{2/3} = \frac{2}{3} (xy)^{-1/3} \cdot y \).Combine these to get: \[ \frac{\partial f}{\partial x} = \frac{3}{2} y (xy)^{1/2} - \frac{2}{3} y (xy)^{-1/3} \].

Step 3: Differentiate with Respect to \( y \)

Next, we find \( \frac{\partial f}{\partial y} \):- The first term for \( y \) differentiation, \( (xy)^{3/2} \), gives: \( \frac{d}{dy} (xy)^{3/2} = \frac{3}{2} (xy)^{1/2} \cdot x \).- The second term, \( (xy)^{2/3} \), gives: \( \frac{d}{dy} (xy)^{2/3} = \frac{2}{3} (xy)^{-1/3} \cdot x \).Thus,\[ \frac{\partial f}{\partial y} = \frac{3}{2} x (xy)^{1/2} - \frac{2}{3} x (xy)^{-1/3} \].

Key Concepts

Understanding the Chain Rule in Calculus

The chain rule is a powerful tool in calculus, and it's especially useful when dealing with composite functions. It helps us find derivatives of functions where one function is nested within another.
When applying the chain rule, you differentiate the outer function and then multiply by the derivative of the inner function. For instance, if you have a function like \( g(x) = (u(x))^n \), where \( u(x) \) is a function of \( x \), the derivative would be \( g'(x) = n
(u(x))^{n-1}
\cdot u'(x) \).

• For functions of more than one variable, the chain rule is applied individually to each variable while treating the others as constants.
• This becomes particularly useful in multivariable calculus, allowing us to explore functions with more complexity and detail.

The chain rule simplifies problems that seem hard at first glance, providing a systematic way to break them down into manageable steps.

Delving into Multivariable Calculus

Multivariable calculus extends the ideas of differentiation and integration from single-variable calculus to functions of several variables. This branch of mathematics is crucial for understanding complex systems in fields such as physics and engineering.

• In multivariable calculus, we often deal with functions that have multiple independent variables like \( f(x, y) \).
• We explore concepts such as partial derivatives and gradients, which help us understand how changes in input affect our outputs.

Partial derivatives are a key element. They show how a function changes with respect to one variable, keeping other variables constant.
This is essential for tasks like:

• Optimizing outputs in manufacturing settings,
• Analyzing temperature changes over an area in meteorology, and
• Predicting outcomes in finance when multiple variables are at play.

Understanding multivariable calculus opens up a world of applications and deeper comprehension of the mathematics that describe our world.

The Basics of Differentiation

Differentiation is a fundamental concept in calculus, focusing on how functions change. It provides us with the derivative, which is a function that describes the rate of change of another function.
For a single-variable function like \( y = f(x) \), differentiation gives us \( f'(x) \), which tells us how \( y \) changes as \( x \) changes.

• Differentiation can be extended into multivariable settings, where we use partial derivatives.
• These are denoted by \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) for functions of two variables \( f(x, y) \).

These help us understand how changes in one variable affect the function's value while keeping others static.
With differentiation, we can:

• Determine the slope of a curve at a point,
• Find rates of change in physical systems, and
• Anlayze the behavior of functions.

Differentiation is a gateway tool, equipping us with insights into both simple and complex interactions between varying quantities.