Final Answer
$\frac{\left(2x\left(x^2+2x-3\right)+\left(x^2-16\right)\left(2x+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(2x+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Step-by-step Solution
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1
Simplifying
$\frac{d}{dx}\left(\frac{\left(x^2-16\right)\left(x^2+2x-3\right)}{\left(x^2+6x+8\right)\left(x-1\right)}\right)$
2
Apply the quotient rule for differentiation, 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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{\frac{d}{dx}\left(\left(x^2-16\right)\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)-\left(x^2-16\right)\left(x^2+2x-3\right)\frac{d}{dx}\left(\left(x^2+6x+8\right)\left(x-1\right)\right)}{\left(\left(x^2+6x+8\right)\left(x-1\right)\right)^2}$
3
The power of a product is equal to the product of it's factors raised to the same power
$\frac{\frac{d}{dx}\left(\left(x^2-16\right)\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)-\left(x^2-16\right)\left(x^2+2x-3\right)\frac{d}{dx}\left(\left(x^2+6x+8\right)\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
4
Simplify the product $-(x^2-16)$
$\frac{\frac{d}{dx}\left(\left(x^2-16\right)\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\frac{d}{dx}\left(\left(x^2+6x+8\right)\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
5
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^2-16$ and $g=x^2+2x-3$
$\frac{\left(\frac{d}{dx}\left(x^2-16\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\frac{d}{dx}\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\frac{d}{dx}\left(\left(x^2+6x+8\right)\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
6
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$, where $f=x^2+6x+8$ and $g=x-1$
$\frac{\left(\frac{d}{dx}\left(x^2-16\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\frac{d}{dx}\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\frac{d}{dx}\left(x^2+6x+8\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
7
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\frac{d}{dx}\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\frac{d}{dx}\left(x^2+6x+8\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
8
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\frac{d}{dx}\left(x^2+6x+8\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
9
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
10
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
11
The derivative of the constant function ($-16$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
12
The derivative of the constant function ($-3$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
13
The derivative of the constant function ($8$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
14
The derivative of the constant function ($-1$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
15
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
16
The derivative of the linear function times a constant, is equal to the constant
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
17
The derivative of the linear function times a constant, is equal to the constant
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+6\frac{d}{dx}\left(x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
18
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+6\frac{d}{dx}\left(x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
19
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
20
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{\left(2x\left(x^2+2x-3\right)+\left(x^2-16\right)\left(2x+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(2x+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
Final Answer
$\frac{\left(2x\left(x^2+2x-3\right)+\left(x^2-16\right)\left(2x+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(2x+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$