Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Find the first partial derivatives with respect to \(x\) and with respect to \(y\). $$ f(x, y)=3 x-6 y^{2} $$

Step-by-Step Solution

Answer

The first partial derivative of \(f\) with respect to \(x\) is 3, and the first partial derivative of \(f\) with respect to \(y\) is \(-12y\).

Step 1: Finding the Partial Derivative with Respect to \(x\)

To find the partial derivative of \(f\) with respect to \(x\) (\(f_x\) or \(\partial f/\partial x\)), treat \(y\) as a constant and differentiate \(f\) with respect to \(x\). For the function \(f(x, y) = 3x - 6y^2\), the derivative of \(3x\) with respect to \(x\) is 3 and the derivative of \(-6y^2\) with respect to \(x\) is 0, since \(y\) is treated as a constant. Therefore, the partial derivative of \(f\) with respect to \(x\) is 3.

Step 2: Finding the Partial Derivative with Respect to \(y\)

To find the partial derivative of \(f\) with respect to \(y\) (\(f_y\) or \(\partial f/\partial y\)), treat \(x\) as a constant and differentiate \(f\) with respect to \(y\). For the function \(f(x, y) = 3x - 6y^2\), the derivative of \(3x\) with respect to \(y\) is 0, since \(x\) is treated as a constant and the derivative of \(-6y^2\) with respect to \(y\) is \(-12y\). Therefore, the partial derivative of \(f\) with respect to \(y\) is \(-12y\).

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus. It involves functions of more than one variable. Unlike single-variable calculus, where you only deal with curves, multivariable calculus allows you to work with surfaces and shapes. This makes it very useful in physics, engineering, and economics.

A function like \( f(x, y) = 3x - 6y^2 \) is an example of a bivariate function, illustrating multivariable calculus. In this function, \(x\) and \(y\) are independent variables. Each pair \((x, y)\) gives a different value of \(f(x, y)\), creating a surface on a three-dimensional plane.

Multivariable calculus helps to analyze and understand complex systems, which involve multiple changing factors, by studying the relationships between these variables.

Differentiation

Differentiation is a fundamental concept in calculus used to determine the rate at which a function is changing. In simple terms, it helps to find the

• instantaneous rate of change
• slope of a function at any given point.

When dealing with functions of more than one variable, differentiation involves taking the derivative with respect to each variable separately.
One variable is treated as changing, while the others are held constant.
This is crucial in examining how each variable influences the function separately, giving insights into the behavior of complex systems. Differentiation in multivariable calculus aids in optimizing functions to find maximum or minimum values and understanding variance in functions.

Partial Derivative with Respect to x

A partial derivative with respect to \(x\) means that we are interested in the rate of change of the function as \(x\) changes, with all other variables held constant. This approach simplifies the differentiation process in multivariable functions.

For the function \(f(x, y) = 3x - 6y^2\), when finding the partial derivative with respect to \(x\), denoted as \(\partial f / \partial x\) or \(f_x\), we treat \(y\) as a constant. The term \(3x\) differentiates to 3, and since \(-6y^2\) is independent of \(x\), its derivative is 0.

This means that no matter what \(y\) value is, the rate of change concerning \(x\) is constant, explained by the expression \(\frac{\partial f}{\partial x} = 3\). This indicates how changes in \(x\) alone affect the function.

Partial Derivative with Respect to y

Partial derivatives with respect to \(y\) focus on how changes in \(y\) affect the function, holding \(x\) constant. This is highlighted in functions like \(f(x, y) = 3x - 6y^2\).

In finding the partial derivative with respect to \(y\), denoted \(\partial f / \partial y\) or \(f_y\), \(x\) is treated as a constant. This results in the derivative of \(3x\) being 0, since it's independent of \(y\). The differentiation of \(-6y^2\) with respect to \(y\) yields \(-12y\).

The expression \(\frac{\partial f}{\partial y} = -12y\) describes how the function's value decreases as \(y\) increases, keeping \(x\) fixed. This kind of analysis helps understand how different variables individually affect a multivariable function.