Problem 1

Question

Find $\partial f / \partial x$ and $\partial f / \partial y$ for the given functions. $f(x, y)=x^{2} y+x y^{2}$

Step-by-Step Solution

Verified

Answer

$\frac{\partial f}{\partial x} = 2xy + y^2, \frac{\partial f}{\partial y} = x^2 + 2xy$.

1Step 1: Understanding Partial Derivatives

Partial derivatives involve differentiating a multivariable function with respect to one variable while keeping the other variable constant. For the function $f(x, y) = x^2 y + xy^2$, we need to find the derivatives with respect to $x$ and $y$.

2Step 2: Finding the Partial Derivative with Respect to x

To find $\frac{\partial f}{\partial x}$, treat $y$ as a constant. Differentiate each term of $f(x, y)$ with respect to $x$. - The derivative of $x^2 y$ with respect to $x$ is $2xy$ because $y$ is a constant.- The derivative of $xy^2$ with respect to $x$ is $y^2$ because $y^2$ is a constant.Thus, $\frac{\partial f}{\partial x} = 2xy + y^2$.

3Step 3: Finding the Partial Derivative with Respect to y

Now, find $\frac{\partial f}{\partial y}$ by treating $x$ as a constant.- The derivative of $x^2 y$ with respect to $y$ is $x^2$ because $x^2$ is constant.- The derivative of $xy^2$ with respect to $y$ is $2xy$ because $x$ is constant and applying the power rule to $y^2$.Thus, $\frac{\partial f}{\partial y} = x^2 + 2xy$.

Key Concepts

Understanding Partial Derivatives in Multivariable CalculusDelving into Differentiation for Multivariable FunctionsApplying the Power Rule in Partial Derivatives

Understanding Partial Derivatives in Multivariable Calculus

Multivariable calculus extends the concepts of calculus into functions with more than one variable. In this field, functions can depend on variables like $x$ and $y$, and one common task is to find how a function changes as these variables change. This is where partial derivatives come into play.

Partial derivatives are specifically used to understand how a multivariable function behaves with respect to one variable at a time, while all other variables are kept constant. Here's how it works in the given function $f(x, y) = x^2y + xy^2$:

To find $\frac{\partial f}{\partial x}$, we vary $x$ and keep $y$ constant.
To find $\frac{\partial f}{\partial y}$, we vary $y$ and keep $x$ constant.

This type of differentiation allows us to understand the sensitivity of the function $f(x, y)$ to each specific variable. Insight into partial derivatives is fundamental for studying changes in surfaces and shapes described by multivariable functions, making it essential in fields like physics and engineering.

Delving into Differentiation for Multivariable Functions

Differentiation in the context of multivariable functions involves different rules while following the same basic principle of finding the derivative of a function. In simple terms, it measures the rate at which something changes.

When working with functions of two or more variables, differentiation is used to find partial derivatives. For the function $f(x, y)$, we follow these steps:

Identify the Variable: Decide which variable the partial derivative should respect, either $x$ or $y$.
Treat Other Variables as Constants: Fix the other variables as constants during the differentiation.
Apply Derivative Rules: Use standard differentiation rules like the power rule, sum rule, and constant multiple rule to find the derivative.

The goal of this process is to observe how the function changes specifically with one variable while the others remain fixed, capturing the influence of a single input on the overall function.

Applying the Power Rule in Partial Derivatives

The power rule is one of the cornerstone rules in calculus used to find derivatives and is very relevant when dealing with multivariable calculus. The power rule states that for a function $x^n$, its derivative is $nx^{n-1}$.

In the context of partial derivatives, this rule can be applied when one variable is treated as constant. For the function $f(x, y) = x^2y + xy^2$, here’s how the power rule is used:

For $\frac{\partial f}{\partial x}$, treat $y$ as a constant and apply the power rule to $x^2y$ to get $2xy$.
Similarly, in $xy^2$, since $y^2$ is constant regarding $x$, the derivative becomes $y^2$.
For $\frac{\partial f}{\partial y}$, treat $x$ as a constant and apply the power rule to $xy^2$ to obtain $2xy$.

Using the power rule effectively simplifies the process of finding partial derivatives, and understanding it deeply aids in tackling more intricate problems in multivariable calculus.

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