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Step-by-step Solution

Find the derivative $\frac{d}{dx}\left(-4x\left(\frac{x}{\left(x^2-8\right)^2}\right)+2\left(\frac{1}{x^2-8}\right)\right)$ using the sum rule

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Answer

$-4\left(x\frac{\left(x^2-8\right)^2-4x^2\left(x^2-8\right)}{\left(x^2-8\right)^{4}}+\frac{x}{\left(x^2-8\right)^2}\right)+\frac{-4x}{\left(x^2-8\right)^2}$

Step-by-step explanation

Problem to solve:

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

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=2$ and $x=x^2-8$

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

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

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

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Answer

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

Main topic:

Sum rule

Used formulas:

7. See formulas

Time to solve it:

~ 0.9 seconds