Problem 25 - Advanced Engineering Mathematics

Question

Find the first partial derivatives of the given function. $$ f(x, y)=\frac{3 x-y}{x+2 y} $$

Step-by-Step Solution

Answer

\( \frac{\partial f}{\partial x} = \frac{7y}{(x + 2y)^2} \), \( \frac{\partial f}{\partial y} = \frac{-7x}{(x + 2y)^2} \).

Step 1: Identify the function and variables

The given function is \( f(x, y) = \frac{3x - y}{x + 2y} \). Here, \( x \) and \( y \) are the variables with respect to which we need to find the partial derivatives.

Step 2: Find the partial derivative with respect to x

To find the partial derivative \( \frac{\partial f}{\partial x} \), treat \( y \) as a constant and differentiate \( f(x, y) = \frac{3x - y}{x + 2y} \) with respect to \( x \). Using the quotient rule \( \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2} \), where \( u = 3x - y \) and \( v = x + 2y \), compute:\[\frac{\partial f}{\partial x} = \frac{(x + 2y)(3) - (3x - y)(1)}{(x + 2y)^2} = \frac{3x + 6y - 3x + y}{(x + 2y)^2} = \frac{7y}{(x + 2y)^2}\]

Step 3: Find the partial derivative with respect to y

To find the partial derivative \( \frac{\partial f}{\partial y} \), treat \( x \) as a constant and differentiate \( f(x, y) = \frac{3x - y}{x + 2y} \) with respect to \( y \). Using the quotient rule again:Let \( u = 3x - y \) and \( v = x + 2y \). Then:\[\frac{\partial f}{\partial y} = \frac{(x + 2y)(-1) - (3x - y)(2)}{(x + 2y)^2}\]Simplify the expression:\[= \frac{-x - 2y - 6x + 2y}{(x + 2y)^2} = \frac{-7x}{(x + 2y)^2}\]

Step 4: Present final results

The first partial derivative with respect to \( x \) is \( \frac{\partial f}{\partial x} = \frac{7y}{(x + 2y)^2} \), and with respect to \( y \) is \( \frac{\partial f}{\partial y} = \frac{-7x}{(x + 2y)^2} \).

Key Concepts

Understanding the Quotient Rule

The quotient rule is a powerful tool in calculus used when differentiating a function that is the division of two other functions. Imagine you have two functions, \( u(x) \) and \( v(x) \), where \( u(x) \) is the numerator and \( v(x) \) is the denominator of your given function \( \frac{u(x)}{v(x)} \). The quotient rule helps us find the derivative of this division.The formula for the quotient rule is:- \[ \frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v \cdot \frac{du}{dx} - u \cdot \frac{dv}{dx}}{v^2} \]This rule states that to find the derivative, you multiply the derivative of the numerator by the denominator, subtract from it the product of the numerator and the derivative of the denominator, and then divide the entire expression by the square of the denominator.In the context of partial derivatives, like in the example above, we apply the quotient rule while treating all other variables as constants during differentiation. This focused approach ensures accuracy when dealing with functions that depend on multiple variables.

Exploring Multivariable Calculus

Multivariable calculus expands on the foundational concepts of calculus by considering functions of more than one variable. While single-variable calculus deals with functions such as \( f(x) \), multivariable calculus examines functions like \( f(x, y) \), where multiple independent variables influence the outcome.In multivariable calculus, differentiation is more nuanced because we must consider how each variable independently affects the function. This leads us to the concept of partial derivatives, which measure the rate of change of the function concerning one variable while keeping the other variables constant. Partial derivatives are crucial in fields such as physics and engineering, where systems often depend on several interacting components.In our example, the task involves finding the partial derivatives of a function \( f(x, y) \), making use of the quotient rule, showcasing how multivariable functions can be analyzed and differentiated to understand their behavior better.

The Basics of Differentiation

Differentiation is one of the core concepts in calculus, used to compute the rate at which a function changes. For single-variable functions, this simply involves finding the derivative with respect to one variable, which gives us a function that represents the slope or rate of change of the original function.When we move into the realm of multivariable calculus, differentiation takes the form of partial derivatives. Unlike ordinary derivatives, partial derivatives involve differentiating a function with respect to one variable while treating all other variables as constants. This allows us to examine how a change in one specific variable impacts the function without being influenced by changes in other variables.In practical terms, whenever you see a function like \( f(x, y) = \frac{3x - y}{x + 2y} \), you're tasked with finding how changing \( x \) or \( y \) independently influences \( f \). Partial derivatives help us break down these dependencies, making them an invaluable tool for understanding complex systems involving multiple variables. In our example, by differentiating partially with respect to each variable using the quotient rule, we gain insight into each variable's specific contribution to changes in the function.