Problem 4 - Calculus with Concepts in Calculus

Question

Find the first partial derivatives of the function. $$ g(x, y)=x^{3} e^{2 y} $$

Step-by-Step Solution

Answer

\( \frac{\partial g}{\partial x} = 3x^2 e^{2y} \), \( \frac{\partial g}{\partial y} = 2x^{3} e^{2y} \).

Step 1: Understand the Function

We have a function \( g(x, y) = x^{3} e^{2y} \). It is a multivariable function with variables \( x \) and \( y \). Our goal is to find its first partial derivatives with respect to \( x \) and \( y \).

Step 2: Partial Derivative with respect to x

To find the partial derivative of \( g \) with respect to \( x \), treat \( y \) as a constant. The derivative of \( x^{3} \) is \( 3x^2 \), and \( e^{2y} \) remains unchanged because it's treated as a constant. Thus, the partial derivative \( \frac{\partial g}{\partial x} \) is \( 3x^2 e^{2y} \).

Step 3: Partial Derivative with respect to y

To find the partial derivative of \( g \) with respect to \( y \), treat \( x \) as a constant. The derivative of \( e^{2y} \) with respect to \( y \) is \( 2e^{2y} \) (using the chain rule). In this case, \( x^{3} \) acts as a constant multiplier, so the partial derivative \( \frac{\partial g}{\partial y} \) is \( 2x^{3} e^{2y} \).

Key Concepts

Multivariable Calculus

Multivariable calculus is an essential branch of mathematics dealing with functions that have more than one variable. These functions, also known as multivariable functions, are not restricted to a single path or direction of change. Instead, they consider multiple directions or dimensions for each variable involved.

In the given exercise, we are working with a function of two variables, expressed as \( g(x, y) = x^3 e^{2y} \). Here, both \(x\) and \(y\) can vary independently, making this a multivariable function. When studying these functions, we often seek to understand how changing each variable individually affects the overall function.

To achieve this, we use partial derivatives, which are a fundamental concept in multivariable calculus. They help us find the rate of change of the function concerning one variable at a time while keeping the other variables constant. This understanding is crucial in various fields like physics, economics, and engineering, where systems often depend on several influencing factors.

Chain Rule

The chain rule is a pivotal method in calculus used to find the derivative of composite functions. In multivariable functions, it helps us compute the derivative of functions where one variable is defined in terms of another.

In the problem here, when calculating the partial derivative of \( g(x, y) = x^3 e^{2y} \) with respect to \(y\), the chain rule comes into play. The function \( e^{2y} \) within our expression is a perfect candidate for applying the chain rule, as it consists of an outer function \( e^u \) and an inner function \( u = 2y \).

The derivative of \( e^u \) is \( e^u \) itself, and the derivative of \( u = 2y \) with respect to \( y \) is 2. By the chain rule, multiplying these gives us the derivative of \( e^{2y} \) with respect to \( y \) as \( 2e^{2y} \). This application of the chain rule significantly simplifies the complexity of finding derivatives in multivariable calculus.

Multivariable Functions

Multivariable functions are a type of function that harnesses two or more input variables to produce an output. These are used to model complex systems where multiple factors influence the outcome. The function \( g(x, y) = x^3 e^{2y} \) is an excellent example, where both \( x \) and \( y \) influence the function's result.

Understanding how each variable individually affects the result helps in predicting behavior or optimizing systems. By taking the partial derivative with respect to \( x \), we hold \( y \) constant and compute how \( g(x, y) \) changes as \( x \) varies. Conversely, taking the partial derivative with respect to \( y \) while holding \( x \) constant shows how changing \( y \) affects the function.

These insights are not only critical in mathematical theory but also in applying practical solutions across disciplines like computer graphics, where surfaces depend on multiple parameters, or in biology, to model systems with various interdependent factors.