Problem 5 - Applied Calculus

Question

Find the partial derivatives in Problems. The variables are restricted to a domain on which the function is defined. $$\frac{\partial z}{\partial x} \text { if } z=x^{2} e^{y}$$

Step-by-Step Solution

Answer

\( \frac{\partial z}{\partial x} = 2x e^y \)

Step 1: Identify the Function and Variables

The given function is \( z = x^2 e^y \). The variables are \( x \) and \( y \). We are asked to find the partial derivative of \( z \) with respect to \( x \).

Step 2: Apply the Definition of Partial Derivative

The partial derivative of \( z \) with respect to \( x \), denoted as \( \frac{\partial z}{\partial x} \), is found by differentiating \( z \) with respect to \( x \) while treating \( y \) as a constant. This means we differentiate \( x^2 e^y \) with respect to \( x \).

Step 3: Differentiate the Function

Since \( y \) is treated as a constant, \( e^y \) is constant with respect to \( x \). Hence, differentiate \( x^2 \cdot e^y \) with respect to \( x \) using the power rule: \( \frac{d}{dx}[x^2] = 2x \). Multiply by the constant \( e^y \), giving \( 2x e^y \).

Step 4: Write the Partial Derivative

The partial derivative of \( z \) with respect to \( x \) is \( \frac{\partial z}{\partial x} = 2x e^y \).

Key Concepts

Multivariable Calculus

Multivariable calculus is a branch of mathematics that extends calculus to functions of multiple variables. While single-variable calculus deals with functions of one variable, multivariable calculus deals with functions that have two or more variables, like \( f(x, y) \) or \( f(x, y, z) \). In this scope, you will often calculate partial derivatives, which are similar to regular derivatives but focus on one variable at a time while keeping the others constant.
This concept is vital in various fields including physics, engineering, and economics where systems are described by multiple varying quantities. By isolating one variable and examining its impact, you obtain a clearer understanding of the system's behavior. In problems involving functions like \( z = x^2 e^y \), identifying the effect of \( x \) while keeping \( y \) unchanged helps derive specific insights.
Multivariable calculus not only generalizes concepts from single-variable calculus but also introduces new ideas like multiple integrals and vector analysis. This makes it crucial for modeling real-world scenarios that encompass a range of factors.

Power Rule

The power rule is an essential technique in calculus for finding derivatives of functions in the form \( x^n \), where \( n \) is any real number. To differentiate \( x^n \), you multiply the exponent \( n \) by \( x^{n-1} \).

• This rule is straightforward: \( \frac{d}{dx} [x^n] = nx^{n-1} \).
• It's one of the first rules learned in calculus due to its simplicity and frequent application.

In the context of partial derivatives and multivariable functions, the power rule remains applicable. For example, to find the partial derivative of \( z = x^2 e^y \) with respect to \( x \), you treat \( e^y \) as a constant and apply the power rule to \( x^2 \).
Understanding and effectively using the power rule enables quick differentiation of polynomial terms, which is a critical skill for tackling more complex mathematical problems.

Exponential Function

The exponential function \( e^y \) is fundamental in mathematics due to its unique properties. One of the most notable properties is that the derivative of \( e^x \) with respect to \( x \) is itself \( e^x \), making it an essential entity in differential equations and growth models.
In multivariable calculus, exponential functions often appear with multiple variables, such as \( z = x^2 e^y \). Here, while taking a partial derivative with respect to \( x \), you treat \( e^y \) as a constant. This simplifies the differentiation process as you focus primarily on the polynomial part involving \( x \), while simply carrying \( e^y \) over in the result.
Recognizing when to hold parts of the expression constant, due to their relation to the variable of differentiation, helps in simplifying complex expressions, ensuring accurate and efficient calculations.