Problem 5 - Calculus an Applied Approach

Question

Find the first partial derivatives with respect to \(x\) and with respect to \(y .\) $$ f(x, y)=\frac{x}{y} $$

Step-by-Step Solution

Answer

The first partial derivative of the function \(f(x, y)=\frac{x}{y}\) with respect to \(x\) is \(\frac{1}{y}\), and with respect to \(y\) is \(-\frac{x}{y^2}\)

Step 1: Partial Derivation with respect to \(x\)

Hold \(y\) as a constant, and differentiate as if \(x\) is the only variable: \(\frac{df}{dx} = \frac{Y \cdot 1 - X \cdot 0}{Y^2} = \frac{1}{y}\) where Y is a constant.

Step 2: Partial Derivation with respect to \(y\)

Hold \(x\) as a constant, and differentiate as if \(y\) is the only variable: \(\frac{df}{dy} = \frac{x \cdot 0 - 1 \cdot y}{(y^2)^2} = \frac{-x}{y^2}\) where X is a constant.

Key Concepts

First Partial Derivative

The first partial derivative is an important concept in multivariable calculus. It involves finding the derivative of a function with respect to one variable while treating all other variables as constants.
In simpler terms, we look at how the function changes when we tweak one variable, assuming the others stay still.

In mathematical analysis, partial derivatives help us understand the gradient or the slope of a surface represented by the function at a particular point. For functions of two variables, like the function given in the exercise, we often consider how changes in one variable affect the output, while the other variable remains unchanged.
This gives us insights into how sensitive the function is to changes in each variable.

• The first partial derivative with respect to a variable examines the rate of change of the function with a small change in that variable.
• It offers a way to measure how a function behaves differently along the different coordinate axes.

Understanding and calculating first partial derivatives are crucial for applications in optimizing multivariable functions and in fields like economics, engineering, and physics.

Differentiation with Respect to x

Differentiation with respect to a variable means finding out how a function changes as that specific variable changes, keeping other variables constant.
With respect to the function given, \(f(x, y)=\frac{x}{y}\), when we differentiate with respect to \(x\), \(y\) is treated like a number, or a constant.

To compute the first partial derivative with respect to \(x\):
- Remember the basic rule of differentiating a constant multiple function, where you treat the constant as multiplying the derivative of the varying part.

• In this case, treating \(y\) as constant, the expression simplifies to just \(\frac{1}{y}\).
• This indicates that for every unit increase in \(x\), the change in \(f(x, y)\) is inversely proportional to \(y\).

This approach helps us pinpoint how graphs of such functions ascend or descend linearly along the \(x\) direction, in relation to the fixed value of \(y\).

Differentiation with Respect to y

Differentiating with respect to \(y\) follows a similar concept as differentiation with respect to \(x\), but here we keep \(x\) constant.
Again considering the function \(f(x, y)=\frac{x}{y}\), focus is taken on \(y\) this time.

For the partial derivative with respect to \(y\): - Differentiating with \(y\) as the variable of interest involves applying the quotient rule since \(y\) is located in the denominator.

• The rule shows that the derivative is \(\frac{-x}{y^2}\).
• This result implies that as \(y\) increases incrementally, the value of \(f(x, y)\) decreases, and the rate of that decrease depends on how large \(x\) is and inversely on \(y\).

This analysis helps us understand changes in terms of how the graph behaves along the \(y\) direction, indicating a reciprocal, dispersing behavior depending on the magnitudes of \(x\) and \(y\).