Problem 3 - Active Calculus Multivariable

Question

Find the partial derivatives of the function $$f(x, y)=x y e^{4 y}$$ \(f_{x}(x, y)=\) ________. \(f_{y}(x, y)=\) _________. \(f_{x y}(x, y)=\) _________. \(f_{y x}(x, y)=\) _________.

Step-by-Step Solution

Answer

\(f_{x}(x, y) = ye^{4y}\). \(f_{y}(x, y) = xe^{4y} + 4x(ye^{4y})\). \(f_{x y}(x, y) = e^{4y} + 4ye^{4y}\). \(f_{y x}(x, y) = e^{4y} + 4ye^{4y}\).

Step 1: Find the partial derivative with respect to x (f_x)

First, let's find the partial derivative of the function f(x, y) with respect to x. To do this, we'll treat y as a constant and differentiate the function with respect to x: \[ f_x(x, y) = \frac{\partial}{\partial x} (xye^{4y}) = ye^{4y} \]

Step 2: Find the partial derivative with respect to y (f_y)

Similarly, let's find the partial derivative of the function f(x, y) with respect to y. This time, we'll treat x as a constant and differentiate the function with respect to y: \[ f_y(x, y) = \frac{\partial}{\partial y} (xye^{4y}) = xe^{4y} + 4x(ye^{4y}) \]

Step 3: Find the mixed partial derivative with respect to x and then y (f_{xy})

Now, we'll find the mixed partial derivative by first taking the partial derivative of \(f_x\) with respect to y: \[ f_{xy}(x, y) = \frac{\partial}{\partial y} (f_x) = \frac{\partial}{\partial y} (ye^{4y}) = e^{4y} + 4ye^{4y} \]

Step 4: Find the mixed partial derivative with respect to y and then x (f_{yx})

Lastly, let's find the other mixed partial derivative by taking the partial derivative of \(f_y\) with respect to x: \[ f_{yx}(x, y) = \frac{\partial}{\partial x} (f_y) = \frac{\partial}{\partial x} (xe^{4y} + 4x(ye^{4y})) = e^{4y} + 4ye^{4y} \] So, the partial derivatives are: - \(f_x(x, y) = ye^{4y}\) - \(f_y(x, y) = xe^{4y} + 4x(ye^{4y})\) - \(f_{xy}(x, y) = e^{4y} + 4ye^{4y}\) - \(f_{yx}(x, y) = e^{4y} + 4ye^{4y}\)

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus that involves functions of several variables. In this realm, we deal with functions like f(x, y), which depend on two or more variables, as opposed to the functions of a single variable we encounter in basic calculus. When working with such functions, we might be interested in how they change as we vary just one of the variables while keeping the others constant. This is where the concept of partial derivatives comes into play.

Partial derivatives represent the rate at which a function changes with respect to one variable, holding the others constant. We denote them with symbols like f_x or f_y, where the subscript represents the variable with respect to which we are differentiating. They are fundamental in determining the slope of the function in a particular direction and are essential in optimizing multivariate functions, understanding slopes and tangent planes, and solving real-world problems that depend on multiple variables.

Mixed Partial Derivative

In the context of multivariable calculus, the mixed partial derivative is a second-order derivative where we differentiate a function with respect to two different variables, one after the other. For instance, if we start with a function f(x, y), we can first differentiate with respect to x to get f_x, and then differentiate this result with respect to y to find f_xy, which is the mixed partial derivative. The key property of mixed partial derivatives for a function with continuous second derivatives is Clairaut's Theorem, which states that the mixed partials are equal regardless of the order of differentiation. This means f_xy will be equal to f_yx, confirming that the function's curvature due to x and then y is the same as the curvature due to y first and then x. This theorem simplifies the analysis of functions as we can confidently interchange the order of differentiation without altering the result.

Exponential Function Differentiation

Exponential functions are widely used in various fields such as physics, finance, and biology due to their properties of growth and decay. Differentiation of exponential functions in calculus is straightforward when the variable is in the exponent. For example, the derivative of e^x with respect to x is e^x itself. However, when dealing with multivariable calculus, we often encounter exponential functions that involve multiple variables.

When faced with such functions, we apply the rules of differentiation with respect to each variable in turn, keeping the others constant, just as we do with polynomial functions. If an exponential function has a coefficient in the exponent, such as 4y in the term e^(4y), this constant is also accounted for in the differentiation process. Hence, the partial derivative of f(x, y) = xye^(4y) with respect to y involves the product rule because both x and y are multiplying the exponential function, leading to terms involving both the exponential function and its derivative after applying the differentiation rules. This highlights the incredible adaptability of exponential functions to the context of multivariable calculus and presents an interesting challenge when solving real-world problems involving complex growth and decay dynamics.