# Find the derivative of x^2y+xy^3*3=(111*-1)/(x^2)

## \frac{d}{dx}\left(x^2y+3x y^3=\frac{111\left(-1\right)}{x^2}\right)

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$2y\cdot x+0+3y^3=\frac{222}{x^{3}}$

## Step by step solution

Problem

$\frac{d}{dx}\left(x^2y+3x y^3=\frac{111\left(-1\right)}{x^2}\right)$
1

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

$\frac{d}{dx}\left(3xy^3+yx^2\right)=\frac{d}{dx}\left(\frac{-111}{x^2}\right)$
2

Applying the quotient rule which states that if $f(x)$ and $g(x)$ are functions and $h(x)$ is the function defined by ${\displaystyle h(x) = \frac{f(x)}{g(x)}}$, where ${g(x) \neq 0}$, then ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

$\frac{d}{dx}\left(3xy^3+yx^2\right)=\frac{111\frac{d}{dx}\left(x^2\right)+x^2\frac{d}{dx}\left(-111\right)}{\left(x^2\right)^2}$
3

The derivative of the constant function is equal to zero

$\frac{d}{dx}\left(3xy^3+yx^2\right)=\frac{111\frac{d}{dx}\left(x^2\right)+0x^2}{\left(x^2\right)^2}$
4

Any expression multiplied by $0$ is equal to $0$

$\frac{d}{dx}\left(3xy^3+yx^2\right)=\frac{111\frac{d}{dx}\left(x^2\right)+0}{\left(x^2\right)^2}$
5

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

$\frac{d}{dx}\left(3xy^3+yx^2\right)=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
6

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

$\frac{d}{dx}\left(3xy^3\right)+\frac{d}{dx}\left(yx^2\right)=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
7

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

$\frac{d}{dx}\left(3xy^3\right)+y\frac{d}{dx}\left(x^2\right)=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
8

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=3x$ and $g=y^3$

$y\frac{d}{dx}\left(x^2\right)+3x\frac{d}{dx}\left(y^3\right)+y^3\frac{d}{dx}\left(3x\right)=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
9

The derivative of the constant function is equal to zero

$y\frac{d}{dx}\left(x^2\right)+0\cdot 3x+y^3\frac{d}{dx}\left(3x\right)=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
10

Any expression multiplied by $0$ is equal to $0$

$y\frac{d}{dx}\left(x^2\right)+0+y^3\frac{d}{dx}\left(3x\right)=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
11

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

$y\frac{d}{dx}\left(x^2\right)+0+3y^3\frac{d}{dx}\left(x\right)=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
12

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

$y\frac{d}{dx}\left(x^2\right)+0+1\cdot 3y^3=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
13

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

$2y\cdot x+0+1\cdot 3y^3=\frac{111\cdot 2x+0}{\left(x^2\right)^2}$
14

Multiply $3$ times $1$

$2y\cdot x+0+3y^3=\frac{222x+0}{\left(x^2\right)^2}$
15

$x+0=x$, where $x$ is any expression

$2y\cdot x+0+3y^3=\frac{222x}{\left(x^2\right)^2}$
16

Applying the power of a power property

$2y\cdot x+0+3y^3=\frac{222x}{x^{4}}$
17

Simplifying the fraction by $x$

$2y\cdot x+0+3y^3=\frac{222}{x^{3}}$

$2y\cdot x+0+3y^3=\frac{222}{x^{3}}$

### Main topic:

Differential calculus

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