Question

Find \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial^{2} f}{\partial x^{2}}, \frac{\partial^{2} f}{\partial y^{2}}\) and \(\frac{\partial^{2} f}{\partial x \partial y}\) if \(f=(x-y)^{2}\).

Short answer

Answer: The first and second order partial derivatives of the function \(f=(x-y)^2\) are as follows: 1. First order partial derivatives: $$ \frac{\partial f}{\partial x}=2(x-y), $$ $$ \frac{\partial f}{\partial y}=-2(x-y), $$ 2. Second order partial derivatives: $$ \frac{\partial^{2} f}{\partial x^{2}}=2, $$ $$ \frac{\partial^{2} f}{\partial y^{2}}=2, $$ $$ \frac{\partial^{2} f}{\partial x \partial y}=-2. $$

Step-by-step solution

(Step 1: Compute the first order partial derivatives)

To find the first order partial derivatives, we differentiate the function \(f=(x-y)^{2}\) with respect to \(x\) and with respect to \(y\). $$ \frac{\partial f}{\partial x} =2(x-y), $$ and $$ \frac{\partial f}{\partial y} =-2(x-y). $$

(Step 2: Compute the second order partial derivatives)

Now we need to differentiate the first order partial derivatives with respect to \(x\) and \(y\) again to find the second order partial derivatives. Differentiating \(\frac{\partial f}{\partial x}\) with respect to \(x\), we get: $$ \frac{\partial^{2} f}{\partial x^{2}} = 2. $$ Differentiating \(\frac{\partial f}{\partial x}\) with respect to \(y\), we get: $$ \frac{\partial^{2} f}{\partial x \partial y} =-\frac{\partial^{2} f}{\partial y \partial x}= -2. $$ Differentiating \(\frac{\partial f}{\partial y}\) with respect to \(y\), we get: $$ \frac{\partial^{2} f}{\partial y^{2}} = 2. $$ So, the first and second order partial derivatives of the function \(f=(x-y)^2\) are: $$ \frac{\partial f}{\partial x}=2(x-y), $$ $$ \frac{\partial f}{\partial y}=-2(x-y), $$ $$ \frac{\partial^{2} f}{\partial x^{2}}=2, $$ $$ \frac{\partial^{2} f}{\partial y^{2}}=2, $$ and $$ \frac{\partial^{2} f}{\partial x \partial y}=-2. $$

Key concepts

Calculus

Calculus is the branch of mathematics that studies continuous change. It consists of two main parts: differential calculus and integral calculus. Differential calculus is concerned with finding how a function changes at any given point, and it primarily involves the concept of derivatives.
Integral calculus, on the other hand, deals with the accumulation of quantities and the areas under and between curves.
Understanding calculus is crucial for gaining deep insights into various scientific fields. It helps in describing motion, growth, and other natural phenomena.

• **Derivative**: Measures how a function changes as its input changes.
• **Integral**: Involves summing infinitely many small quantities.

In context with the solution exercise, we are mostly dealing with the concept of derivatives in calculus.

First Order Derivative

The first order derivative of a function provides information about the rate at which the function's value is changing. For a function of one variable, it gives the slope of the tangent to the curve of the function at any point.
For functions of two variables, like in our exercise, the first order partial derivatives tell us how the function changes as either variable changes, while keeping the other variable constant.
For the function \(f=(x-y)^2\), the first order partial derivatives are:

• \(\frac{\partial f}{\partial x} = 2(x-y)\): This derivative measures the change in the function as \(x\) changes, while \(y\) remains constant.
• \(\frac{\partial f}{\partial y} = -2(x-y)\): This derivative measures the change in the function as \(y\) changes, while \(x\) remains constant.

These derivatives indicate the local rate of change in different directions.

Second Order Derivative

Second order derivatives provide further analysis into the curvature and concavity of functions. They reflect how the rate of change itself is changing.
In the context of two variables, second order partial derivatives involve taking the derivative of a first order partial derivative.
For our task function \(f=(x-y)^2\), these derivatives are:

• \(\frac{\partial^2 f}{\partial x^2} = 2\)
• \(\frac{\partial^2 f}{\partial y^2} = 2\)
• Mixed derivative: \(\frac{\partial^2 f}{\partial x \partial y} = -2\)

The second order derivatives help understand the behavior of the function at a more detailed level. For instance, a positive second order derivative indicates that the function is concave up, contributing to seeing whether points are maxima, minima, or saddle points.

Function of Two Variables

A function of two variables depends on two different inputs, typically denoted as \(x\) and \(y\). These functions map pairs of inputs to a single output value.
They can be used to model a wide range of real-world phenomena, such as temperature distribution over an area or altitude surfaces.The exercise given is based on such a function: \(f=(x-y)^2\).
The analysis of functions of two variables often involves exploring how the function behaves as each input changes independently. This involves the use of partial derivatives to understand the sensitivity of the function with respect to each variable.
Partial derivatives give us insight into the slope and curvature of these functions experienced in multiple dimensions, crucial for graphing surfaces and optimizing these functions.