Problem 39 - Calculus and its applications

Question

Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\). $$f(x, y)=y \ln x$$

Step-by-Step Solution

Answer

The second-order partial derivatives are: \(f_{xx} = -\frac{y}{x^2}\), \(f_{xy} = \frac{1}{x}\), \(f_{yx} = \frac{1}{x}\), \(f_{yy} = 0\).

Step 1: Find the First Partial Derivative with respect to x

To find the partial derivative of the function with respect to \(x\), you treat \(y\) as a constant. For \(f(x, y) = y \ln x\), using the derivative \(\frac{d}{dx}(\ln x) = \frac{1}{x}\), we have:\[ f_x = \frac{\partial}{\partial x} (y \ln x) = y \cdot \frac{1}{x} = \frac{y}{x}. \]

Step 2: Find the First Partial Derivative with respect to y

To find the partial derivative of the function with respect to \(y\), treat \(x\) as a constant. For \(f(x, y) = y \ln x\), the derivative of a constant times a variable is the constant itself, hence:\[ f_y = \frac{\partial}{\partial y} (y \ln x) = \ln x. \]

Step 3: Compute the Second Partial Derivative \(f_{xx}\)

Now, differentiate \(f_x = \frac{y}{x}\) with respect to \(x\) again:We have \(\frac{d}{dx}\left(\frac{y}{x}\right) = -\frac{y}{x^2}\) using the power rule and the fact that \(y\) is still a constant.Thus, \[ f_{xx} = -\frac{y}{x^2}. \]

Step 4: Compute the Mixed Partial Derivative \(f_{xy}\)

Differentiate \(f_x = \frac{y}{x}\) with respect to \(y\):Since \(\frac{y}{x}\) as a function of \(y\) is essentially \(y \cdot \frac{1}{x}\), its derivative is \(\frac{1}{x}\) because \(y\) is treated as the variable.Thus, \[ f_{xy} = \frac{1}{x}. \]

Step 5: Compute the Mixed Partial Derivative \(f_{yx}\)

Now, differentiate \(f_y = \ln x\) with respect to \(x\):The derivative of \(\ln x\) with respect to \(x\) is \(\frac{1}{x}\), so this gives us:\[ f_{yx} = \frac{1}{x}. \]

Step 6: Compute the Second Partial Derivative \(f_{yy}\)

Now, differentiate \(f_y = \ln x\) again, but this time with respect to \(y\):Since \(\ln x\) is independent of \(y\), its derivative with respect to \(y\) is simply 0.Thus, \[ f_{yy} = 0. \]

Key Concepts

Partial Derivatives

Partial derivatives are a core concept in multivariable calculus, essential for functions with more than one variable. Unlike regular derivatives, which involve only one independent variable, partial derivatives show how a multivariable function changes when one specific variable is varied while holding the others constant. This concept is crucial for analyzing how functions behave in a multi-dimensional space.

To compute a partial derivative, you take the derivative of the function with respect to one variable and treat all other variables as constants. For example, in the function given, \(f(x, y) = y \ln x\), the first partial derivative of \(f\) with respect to \(x\) treats \(y\) as constant, resulting in \(f_x = \frac{y}{x}\). This explains how \(f\) changes as \(x\) varies with \(y\) held steady.

Similarly, the derivative with respect to \(y\) treats \(x\) as a constant. For our function, this secondary derivative is simply \(\ln x\). Here, \(f_y = \ln x\) indicates how \(f\) changes as \(y\) alone changes. Partial derivatives allow us to dissect the behavior of complex functions variable by variable, opening up deeper insights into changes and rates across their parameter space.

Second Order Partial Derivatives

Once the first partial derivatives are identified, the journey into the second order begins. In second-order derivatives, you further differentiate the first partial derivatives. These derivatives offer insights into the curvature and concavity of a function's graph, revealing how the rate of change itself is changing.

Consider the function \(f(x, y) = y \ln x\). We determine \(f_{xx}\) by differentiating \(f_x = \frac{y}{x}\) again with respect to \(x\). This gives us \(f_{xx} = -\frac{y}{x^2}\), indicating the concavity concerning \(x\), while \(y\) is constant. This downward curvature reflects how steeply \(f\) decreases as \(x\) continues to rise.

Additionally, calculating \(f_{yy}\) involves differentiating \(f_y = \ln x\) with respect to \(y\), giving us 0, given \(\ln x\) doesn't change with \(y\). Therefore, \(f_{yy} = 0\) shows there's no curvature dependency on \(y\) alone. Second order partial derivatives, therefore, paint a fuller picture regarding the twists and turns of a function's contour.

Mixed Partial Derivatives

Mixed partial derivatives are a fascinating aspect of multivariable calculus. They involve differentiating a multivariable function with respect to two different variables successively. These derivatives examine how a function changes if varying two different dimensions sequentially.

For the function \(f(x, y) = y \ln x\), the mixed partial derivative \(f_{xy}\) is found by differentiating \(f_x = \frac{y}{x}\) with respect to \(y\), leading to \(f_{xy} = \frac{1}{x}\). Meanwhile, \(f_{yx}\) is derived by differentiating \(f_y = \ln x\) with respect to \(x\), also resulting in \(f_{yx} = \frac{1}{x}\). Notably, these mixed partial derivatives equal each other, demonstrating the symmetry property called Schwarz's theorem for most well-behaved functions.

This symmetry has significant implications, simplifying the mathematical analysis of functions. Mixed partial derivatives help in gauging interactions between variables, reflecting how the function changes as one variable affects the others in a multivariable environment. Understanding these derivatives deepens our grasp of function dynamics across different directions in a coordinate system.