Problem 34 - Calculus and its applications

Question

Find \(f_{x x}, f_{x y}, f_{y x},\) and \(f_{y y}\). $$f(x, y)=3 x+5 y$$

Step-by-Step Solution

Answer

All second partial derivatives \(f_{xx}, f_{xy}, f_{yx},\) and \(f_{yy}\) are 0.

Step 1: Understanding Partial Derivatives

The function provided is a multivariable function, and we need to calculate the second-order partial derivatives with respect to the variables \(x\) and \(y\). The notation \(f_{xx}\) denotes the second partial derivative of \(f\) with respect to \(x\) twice. Similarly, \(f_{xy}\) denotes taking the derivative with respect to \(x\) first and then \(y\), \(f_{yx}\) denotes the derivative with respect to \(y\) first and then \(x\), and \(f_{yy}\) denotes the second partial derivative with respect to \(y\) twice.

Step 2: Calculating the First Partial Derivatives

First, find the partial derivative of \(f\) with respect to \(x\): \[ \frac{\partial f}{\partial x} = 3 \]Secondly, find the partial derivative of \(f\) with respect to \(y\):\[ \frac{\partial f}{\partial y} = 5 \]

Step 3: Calculating Second Partial Derivatives with Respect to x

Find \(f_{xx}\), the second partial derivative with respect to \(x\) using \(\frac{\partial f}{\partial x} = 3\):\[ f_{xx} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial}{\partial x} (3) = 0 \]

Step 4: Calculating Mixed Partial Derivative f_{xy}

Find \(f_{xy}\), the derivative with respect to \(x\) then \(y\) using \(\frac{\partial f}{\partial x} = 3\):\[ f_{xy} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = \frac{\partial}{\partial y} (3) = 0 \]

Step 5: Calculating Mixed Partial Derivative f_{yx}

Find \(f_{yx}\), the derivative with respect to \(y\) then \(x\) using \(\frac{\partial f}{\partial y} = 5\):\[ f_{yx} = \frac{\partial}{\partial x} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial}{\partial x} (5) = 0 \]

Step 6: Calculating Second Partial Derivatives with Respect to y

Find \(f_{yy}\), the second partial derivative with respect to \(y\) using \(\frac{\partial f}{\partial y} = 5\):\[ f_{yy} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial y} \right) = \frac{\partial}{\partial y} (5) = 0 \]

Key Concepts

Second-Order Partial Derivatives

The concept of second-order partial derivatives involves taking partial derivatives of a function not once, but twice. Imagine you have a function of two variables, say \(f(x, y)\). Here, you can either take partial derivatives twice with respect to the same variable or with different ones.
The specific notation tells you how to proceed:

• \(f_{xx}\) means take the derivative with respect to \(x\) twice.
• \(f_{yy}\) means to take the derivative with respect to \(y\) twice.

Executing this procedure often helps in understanding the curvature or concavity in a multidimensional context, much like the second derivative in single-variable calculus gives insights into concavity and points of inflection. The values of these derivatives indicate how the function behaves in response to small changes in each variable, offering a robust approach to analyzing multivariable functions.

Multivariable Function

An essential concept in multivariable calculus, a multivariable function is a function that depends on more than one variable. Unlike single-variable functions which can be plotted in two dimensions, multivariable functions exist in a higher-dimensional space. For example, the function in question \(f(x, y) = 3x + 5y\) is a simple two-variable function with linearity.
Key characteristics of multivariable functions include:

• They form surfaces when graphed, with values changing based on multiple inputs.
• They allow for examination of interactions between variables, providing a broad perspective on how systems behave.
• Partial derivatives have to be used to explore changes in these functions, as they tell you how the function changes with respect to one of the variables while holding the others constant.

Understanding multivariable functions is fundamental to fields like physics, economics, and engineering where systems are usually dependent on various variables.

Mixed Partial Derivatives

In the exploration of multivariable functions, mixed partial derivatives are imperative. These derivatives measure the change in slope of a function with respect to two variables. Considering a function \(f(x, y)\), mixed partial derivatives take partial derivatives with respect to different variables.

• \(f_{xy}\) involves differentiating first with respect to \(x\) and then \(y\).
• \(f_{yx}\) is calculated by differentiating first with respect to \(y\) and then \(x\).

The equality \(f_{xy} = f_{yx}\) often holds true for most well-behaved functions due to Clairaut's Theorem, assuming the function is continuous and has continuous partial derivatives. This symmetry implies that the order of differentiation does not matter, allowing flexibility in calculations and often simplifying the computational process. For linear functions like \(f(x, y) = 3x + 5y\), mixed partial derivatives tend to zero because linear functions do not have curvature.