# Derive the function (x(x+4)*-2)/((-1x-2+x^2)^2) with respect to x

## \frac{d}{dx}\left(\frac{-2x\cdot\left(x+4\right)}{\left(x^2-x-2\right)^2}\right)

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$\frac{4x\left(4+x\right)\left(2x-1\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(4+2x\right)}{\left(-2-x+x^2\right)^{4}}$

## Step by step solution

Problem

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

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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$

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

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

$\frac{2x\left(4+x\right)\frac{d}{dx}\left(\left(-2-x+x^2\right)^2\right)-2\left(-2-x+x^2\right)^2\frac{d}{dx}\left(x\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
3

Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x$ and $g=4+x$

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

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

$\frac{2x\left(4+x\right)\frac{d}{dx}\left(\left(-2-x+x^2\right)^2\right)-2\left(-2-x+x^2\right)^2\left(x\frac{d}{dx}\left(4+x\right)+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
5

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

$\frac{2\cdot 2x\left(4+x\right)\left(-2-x+x^2\right)\frac{d}{dx}\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(x\frac{d}{dx}\left(4+x\right)+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
6

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

$\frac{2\cdot 2x\left(4+x\right)\left(-2-x+x^2\right)\left(\frac{d}{dx}\left(-2\right)+\frac{d}{dx}\left(-x\right)+\frac{d}{dx}\left(x^2\right)\right)-2\left(-2-x+x^2\right)^2\left(x\left(\frac{d}{dx}\left(4\right)+\frac{d}{dx}\left(x\right)\right)+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
7

The derivative of the constant function is equal to zero

$\frac{2\cdot 2x\left(4+x\right)\left(-2-x+x^2\right)\left(0+\frac{d}{dx}\left(-x\right)+\frac{d}{dx}\left(x^2\right)\right)-2\left(-2-x+x^2\right)^2\left(x\left(0+\frac{d}{dx}\left(x\right)\right)+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
8

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

$\frac{2\cdot 2x\left(4+x\right)\left(-2-x+x^2\right)\left(0+\frac{d}{dx}\left(-x\right)+\frac{d}{dx}\left(x^2\right)\right)-2\left(-2-x+x^2\right)^2\left(\left(0+1\right)x+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
9

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

$\frac{2\cdot 2x\left(4+x\right)\left(-2-x+x^2\right)\left(0-\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(x^2\right)\right)-2\left(-2-x+x^2\right)^2\left(\left(0+1\right)x+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
10

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

$\frac{2\cdot 2x\left(4+x\right)\left(-2-x+x^2\right)\left(0+1\left(-1\right)+\frac{d}{dx}\left(x^2\right)\right)-2\left(-2-x+x^2\right)^2\left(\left(0+1\right)x+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
11

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

$\frac{2\cdot 2x\left(4+x\right)\left(0+1\left(-1\right)+2x\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(\left(0+1\right)x+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
12

Add the values $1$ and $0$

$\frac{2\cdot 2x\left(4+x\right)\left(0+1\left(-1\right)+2x\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(1x+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
13

Multiply $-1$ times $1$

$\frac{4x\left(4+x\right)\left(0-1+2x\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(1x+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
14

Subtract the values $0$ and $-1$

$\frac{4x\left(4+x\right)\left(2x-1\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(1x+1\left(4+x\right)\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
15

Any expression multiplied by $1$ is equal to itself

$\frac{4x\left(4+x\right)\left(2x-1\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(x+4+x\right)}{\left(\left(-2-x+x^2\right)^2\right)^2}$
16

Applying the power of a power property

$\frac{4x\left(4+x\right)\left(2x-1\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(x+4+x\right)}{\left(-2-x+x^2\right)^{4}}$
17

Adding $x$ and $x$

$\frac{4x\left(4+x\right)\left(2x-1\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(4+2x\right)}{\left(-2-x+x^2\right)^{4}}$

$\frac{4x\left(4+x\right)\left(2x-1\right)\left(-2-x+x^2\right)-2\left(-2-x+x^2\right)^2\left(4+2x\right)}{\left(-2-x+x^2\right)^{4}}$

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Differential calculus

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