# Find the derivative of xy^2+ln(x+y)=0

## \frac{d}{dx}\left(x y^2+\ln\left(x+y\right)=0\right)

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$\frac{1}{y+x}+y^2=0$

## Step by step solution

Problem

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

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

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

The derivative of the constant function is equal to zero

$\frac{d}{dx}\left(\ln\left(y+x\right)+xy^2\right)=0$
3

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

$\frac{d}{dx}\left(\ln\left(y+x\right)\right)+\frac{d}{dx}\left(xy^2\right)=0$
4

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

$\frac{d}{dx}\left(\ln\left(y+x\right)\right)+y^2\frac{d}{dx}\left(x\right)=0$
5

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

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

The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$

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

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

$\frac{1}{y+x}\left(\frac{d}{dx}\left(y\right)+\frac{d}{dx}\left(x\right)\right)+1y^2=0$
8

The derivative of the constant function is equal to zero

$\frac{1}{y+x}\left(0+\frac{d}{dx}\left(x\right)\right)+1y^2=0$
9

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

$\left(0+1\right)\left(\frac{1}{y+x}\right)+1y^2=0$
10

Add the values $1$ and $0$

$1\left(\frac{1}{y+x}\right)+1y^2=0$
11

Any expression multiplied by $1$ is equal to itself

$\frac{1}{y+x}+y^2=0$

$\frac{1}{y+x}+y^2=0$

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### Main topic:

Differential calculus

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