# Derive the function x+y^2+y+x^2=xy with respect to x

## \frac{d}{dx}\left(x^2+y+x+y^2=xy\right)

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$2x+1=y$

## Step by step solution

Problem

$\frac{d}{dx}\left(x^2+y+x+y^2=xy\right)$
1

Apply implicit differentiation by taking the derivative of both sides of the equation with respect to the differentiation variable

$\frac{d}{dx}\left(y^2+x+y+x^2\right)=\frac{d}{dx}\left(y\cdot x\right)$
2

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(y^2+x+y+x^2\right)=y\frac{d}{dx}\left(x\right)$
3

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(y^2+x+y+x^2\right)=1y$
4

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(y^2\right)+\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(y\right)+\frac{d}{dx}\left(x^2\right)=1y$
5

The derivative of the constant function is equal to zero

$0+\frac{d}{dx}\left(x\right)+0+\frac{d}{dx}\left(x^2\right)=1y$
6

The derivative of the linear function is equal to $1$

$0+1+0+\frac{d}{dx}\left(x^2\right)=1y$
7

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$0+1+0+2x=1y$
8

Add the values $1$ and $0$

$2x+1=1y$
9

Any expression multiplied by $1$ is equal to itself

$2x+1=y$

$2x+1=y$

### Main topic:

Differential calculus

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