Problem 19 - Calculus and its applications

Question

Find \(f_{x}\) and \(f_{y}\). $$f(x, y)=4(3 x+y-8)^{2}$$

Step-by-Step Solution

Answer

\( f_{x} = 24(3x + y - 8) \), \( f_{y} = 8(3x + y - 8) \)

Step 1: Identify the partial derivative with respect to x

To find \( f_{x} \), we will differentiate \( f(x, y) = 4(3x + y - 8)^2 \) with respect to \( x \). Use the chain rule, as the function is of the form \( u^2 \) where \( u = 3x + y - 8 \). The derivative of \( u^2 \) with respect to \( u \) is \( 2u \).

Step 2: Apply the chain rule for f_x

According to the chain rule, the derivative of \( f \) with respect to \( x \) is \( f_{x} = 4 \cdot 2(3x + y - 8) \cdot \frac{d}{dx}(3x + y - 8) \). Since \( \frac{d}{dx}(3x + y - 8) = 3 \), we substitute back to get: \[ f_{x} = 8(3x + y - 8) \cdot 3 = 24(3x + y - 8) \].

Step 3: Identify the partial derivative with respect to y

To find \( f_{y} \), differentiate \( f(x, y) = 4(3x + y - 8)^2 \) with respect to \( y \). Again, use the chain rule by noting \( u = 3x + y - 8 \) and \( u^2 \). The derivative of \( u^2 \) with respect to \( u \) is \( 2u \).

Step 4: Apply the chain rule for f_y

Using the chain rule, the derivative of \( f \) with respect to \( y \) is \( f_{y} = 4 \cdot 2(3x + y - 8) \cdot \frac{d}{dy}(3x + y - 8) \). Since \( \frac{d}{dy}(3x + y - 8) = 1 \), this yields: \[ f_{y} = 8(3x + y - 8) \].

Key Concepts

Understanding Calculus

Calculus is the branch of mathematics that studies how things change. At its core, it helps us understand the rate of change and the accumulation of quantities. There are two main branches of calculus:

• Differential Calculus: Focuses on the concept of a derivative, which represents how a function changes as its input changes. It deals with finding the slope of curves.
• Integral Calculus: Deals with finding the total accumulation of quantities, such as areas under curves.

In problems involving functions of multiple variables like in partial derivatives, calculus becomes crucial. Each variable can influence the outcome, and understanding derivatives for each one allows us to see the role they play in shaping the function's behavior.
The exercise above requires us to evaluate how the function changes concerning each variable separately, which is a staple procedure in calculus.

Decoding the Chain Rule

The chain rule is a fundamental technique in differentiation used when dealing with composite functions. A composite function is essentially a combination where a function is applied within another function. When you need to differentiate such functions, the chain rule breaks it down into manageable parts.
For example, in the given function, which is in the form of \((u(x, y))^2\), the chain rule allows us to first take the derivative of the outer function \(u^2\) with respect to \(u\), giving us \(2u\).
Then you multiply by the derivative of the inner function \(u(x,y)\) with respect to the variable of interest.
This rule essentially provides a method to "chain" the differentiation steps in an orderly manner, ensuring accurate outcomes when tackling multivariable problems. Understanding and applying the chain rule correctly is key when finding partial derivatives.

Differentiation in Multivariable Calculus

Differentiation is a central concept in calculus, where we determine how a function changes as its inputs change. In multivariable calculus, we often differentiate functions with more than one variable, which leads to the use of partial derivatives.
A partial derivative involves taking the derivative of the function with respect to one variable while treating the other variables as constants. This highlights the influence that each variable independently exerts on the function’s overall behavior.
In the exercise, to find \(f_x\), the partial derivative of the function with respect to \(x\), we treat \(y\) as constant. Similarly, for \(f_y\), \(x\) is constant.
Such differentiation allows us to accurately determine the rate at which a function changes due to alterations in just one of its multiple variables. Understanding this process is essential for analyzing functions in physics, engineering, economics, and beyond.