Problem 23 - Calculus for Biology and Medicine

Question

Find the indicated partial derivatives. \(f(x, y)=\frac{x y}{x^{2}+2} ; f_{x}(-1,2)\)

Step-by-Step Solution

Answer

The partial derivative \( f_x(-1, 2) \) is \( \frac{2}{9} \).

Step 1: Identify the function and variable

The function is given by \[ f(x, y) = \frac{x y}{x^2 + 2}. \] We need to find the partial derivative with respect to \(x\), which is denoted as \( f_x(x, y) \).

Step 2: Differentiate with respect to x

Use the quotient rule for differentiation, which is:\[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}, \]where \( u = xy \) and \( v = x^2 + 2 \). Compute each part:- \( \frac{du}{dx} = y \)- \( \frac{dv}{dx} = 2x \).The partial derivative \( f_x(x, y) \) is:\[ f_x(x, y) = \frac{(x^2 + 2)y - xy(2x)}{(x^2 + 2)^2}. \]

Step 3: Simplify the expression

Simplify the expression obtained for \( f_x(x, y) \):\[ f_x(x, y) = \frac{yx^2 + 2y - 2x^2y}{(x^2 + 2)^2}. \]Combine like terms to get:\[ f_x(x, y) = \frac{2y - x^2y}{(x^2 + 2)^2}. \]Factor out \( y \):\[ f_x(x, y) = \frac{y(2 - x^2)}{(x^2 + 2)^2}. \]

Step 4: Evaluate at the given point

Now, substitute \( x = -1 \) and \( y = 2 \) into the simplified derivative:\[ f_x(-1, 2) = \frac{2(2 - (-1)^2)}{((-1)^2 + 2)^2}. \]Calculate the numerical values:- Find \( (-1)^2 = 1 \).- Now substitute to find:\[ f_x(-1, 2) = \frac{2(2 - 1)}{(1 + 2)^2} = \frac{2 \times 1}{3^2} = \frac{2}{9}. \]

Key Concepts

quotient rule

The quotient rule is a fundamental technique in calculus used to find the derivative of a function that is the division of two other functions. Specifically, if you have a function defined as \( f(x) = \frac{u(x)}{v(x)} \), where both \( u \) and \( v \) are differentiable functions of \( x \), then the derivative \( f'(x) \) is given by:\[ f'(x) = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{(v(x))^2} \]This rule is incredibly useful for handling ratios of functions, ensuring that even when functions appear complex, we can systematically compute their derivatives.

Let's break down how it applies to our function \( f(x, y) = \frac{xy}{x^2 + 2} \). Here:

• \( u(x) = xy \)
• \( v(x) = x^2 + 2 \)

We then calculate the necessary derivatives:

• \( u'(x) = y \)
• \( v'(x) = 2x \)

By substituting these into the quotient rule formula, we can effectively find \( f_x(x, y) \), the partial derivative of \( f \) with respect to \( x \). This shows the power and utility of the quotient rule in simplifying the process of differentiation.

differentiation

Differentiation is the process of finding a derivative, which is a measure of how a function changes as its input changes. It is a fundamental aspect of calculus, playing a crucial role in understanding the behavior of functions.
Particularly, when working with functions of more than one variable, partial differentiation is used. Here, we focus on one variable while keeping the others constant. This is indicated by notations like \( f_x \) or \( f_y \), showing the derivative of function \( f \) with respect to \( x \) or \( y \) respectively.
In our function, \( f(x, y) = \frac{xy}{x^2 + 2} \), differentiation with respect to \( x \) involves treating \( y \) as a constant. By applying standard differentiation rules like the quotient rule, we can find how \( f \) changes with \( x \). Differentiation allows us to explore the rate of change of surfaces and curves formed by such multidimensional functions, providing deeper insights into their behavior.

functions of multiple variables

Functions of multiple variables extend the idea of a function from mappings of a single input to an output, to multiple inputs. These functions can be visualized as surfaces in three-dimensional spaces, where each point on the surface corresponds to a unique set of inputs.

For example, consider \( f(x, y) = \frac{xy}{x^2 + 2} \). This function takes two inputs \( x \) and \( y \), and calculates an output based on these values. Functions like these are essential in physics, engineering, and economics to model phenomena that depend on more than one factor.
In such functions, finding partial derivatives helps determine how these surfaces tilt or curve with respect to each input variable. This provides valuable information about how small changes in one variable impact the overall function. Whether it's to optimize a shape, model a process, or simply understand the interaction between variables, handling functions of multiple variables is a critical skill in mathematical problem-solving.