Final answer to the problem
$\frac{\left(2x\left(x^2+2x-3\right)+\left(x^2-16\right)\left(2x+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(2x+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Step-by-step Solution
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Find the derivative using the quotient rule Find the derivative using the definition Find the derivative using the product rule Find the derivative using logarithmic differentiation Find the derivative Integrate by partial fractions Product of Binomials with Common Term FOIL Method Integrate by substitution Integrate by parts
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Intermediate steps
$\frac{d}{dx}\left(\frac{\left(x^2-16\right)\left(x^2+2x-3\right)}{\left(x^2+6x+8\right)\left(x-1\right)}\right)$
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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 $-(x^2-16)$
$\frac{\frac{d}{dx}\left(\left(x^2-16\right)\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\frac{d}{dx}\left(\left(x^2+6x+8\right)\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
Intermediate steps
5
Apply the product rule for differentiation: $(f\cdot g)'=f'\cdot g+f\cdot g'$
$\frac{\left(\frac{d}{dx}\left(x^2-16\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\frac{d}{dx}\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\frac{d}{dx}\left(x^2+6x+8\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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6
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\frac{d}{dx}\left(x^2+2x-3\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\frac{d}{dx}\left(x^2+6x+8\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
7
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\frac{d}{dx}\left(x^2+6x+8\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
8
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x-1\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
9
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-16\right)\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
10
The derivative of the constant function ($-16$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(-3\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
11
The derivative of the constant function ($-3$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(8\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
12
The derivative of the constant function ($8$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\left(\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-1\right)\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
13
The derivative of the constant function ($-1$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+\left(x^2+6x+8\right)\frac{d}{dx}\left(x\right)\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
Intermediate steps
14
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(2x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Intermediate steps
15
The derivative of the linear function times a constant, is equal to the constant
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(6x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Intermediate steps
16
The derivative of the linear function times a constant, is equal to the constant
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\frac{d}{dx}\left(x\right)\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+6\frac{d}{dx}\left(x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Intermediate steps
17
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+6\frac{d}{dx}\left(x\right)\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Intermediate steps
18
The derivative of the linear function is equal to $1$
$\frac{\left(\frac{d}{dx}\left(x^2\right)\left(x^2+2x-3\right)+\left(x^2-16\right)\left(\frac{d}{dx}\left(x^2\right)+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(\frac{d}{dx}\left(x^2\right)+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Intermediate steps
19
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{\left(2x\left(x^2+2x-3\right)+\left(x^2-16\right)\left(2x+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(2x+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$
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Final answer to the problem
$\frac{\left(2x\left(x^2+2x-3\right)+\left(x^2-16\right)\left(2x+2\right)\right)\left(x^2+6x+8\right)\left(x-1\right)+\left(-x^2+16\right)\left(x^2+2x-3\right)\left(\left(2x+6\right)\left(x-1\right)+x^2+6x+8\right)}{\left(x^2+6x+8\right)^2\left(x-1\right)^2}$