Problem 9 - Calculus an Applied Approach

Question

Find the first partial derivatives with respect to \(x\) and with respect to \(y .\) $$ z=x^{2} e^{2 y} $$

Step-by-Step Solution

Answer

The first partial derivative of the given function with respect to x is \(2x e^{2y}\), and with respect to y is \(2x^{2} e^{2y}\)

Step 1: Find the partial derivative with respect to x

We start off by finding the derivative with respect to x. This involves only differentiating the parts that involve x, which gives us \(2x e^{2y}\).

Step 2: Find the partial derivative with respect to y

Next, find the derivative with respect to y. This involves only differentiating the parts that involve y, using the product rule, which states that the derivative of a product of two functions is the derivative of the first times the second plus the first times the derivative of the second. This gives us \(2x^{2} e^{2y}\).

Key Concepts

Multivariable Calculus

When delving into the realm of multivariable calculus, one explores functions that have more than one independent variable. In the context of our example where the function is z = x^{2} e^{2y}, both x and y serve as independent variables, and z is the dependent variable. The goal here is to understand how z changes as either x, y, or both change.

The first partial derivatives with respect to x and y provide us with a way to see how z changes in relation to small changes in each variable. It's crucial to visualize this process as slicing the surface defined by z along the x-axis and y-axis. Each slice gives us a curve, and the slope of these curves at a point gives us the partial derivatives at that point.

Product Rule

The product rule is an essential tool in calculus whenever you're faced with differentiating a product of two functions. It states that the derivative of a product u(x)v(x) is given by u'(x)v(x) + u(x)v'(x).

In the context of the given exercise, we had the product of x^{2} and e^{2y}, and when finding the partial derivative with respect to y, we treated x^{2} as a constant and applied the product rule to differentiate e^{2y}. This rule is pivotal as it allows us to break down complex expressions into simpler components that can be differentiated independently.

Exponential Functions Differentiation

Exponential functions are those in which the variable appears in the exponent. Differentiating these functions typically requires an understanding of the exponential function e^x, which is unique because its derivative is itself. For exponential functions of the form e^{f(x)}, where f(x) is a function of x, we apply the chain rule.

In our problem, the function e^{2y} needs to be differentiated with respect to y. The derivative of e^{2y} with respect to y is 2e^{2y} because we multiply the original function by the derivative of the exponent (in this case, 2), following the general differentiation principle for exponential functions.

Chain Rule

The chain rule is a fundamental principle in calculus that's applied when differentiating composite functions - functions made up of two or more other functions. Simply put, if you have a function h(x) = f(g(x)), then the derivative h'(x) is f'(g(x))·g'(x). This means you take the derivative of the outer function and multiply it by the derivative of the inner function.

Our exercise includes an implied use of the chain rule during the differentiation of e^{2y} with respect to y. Here, e^{2y} can be seen as an outer function e^u, where u = 2y is the inner function. Thus, we differentiate e^u to get e^u and multiply it by the derivative of u, which is 2, resulting in 2e^{2y} as the final derivative.