Final answer to the problem
$\frac{6x+3x^{6}+5x^{4}-2x^{5}-4x^{3}}{\left(x^2+1\right)^2}$
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Step-by-step Solution
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Find the derivative Find the derivative using the product rule Find the derivative using the quotient rule Logarithmic Differentiation Find the derivative using the definition Suggest another method or feature
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1
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)-\left(x^5-x^4+x^2-2\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
2
Simplify the product $-(x^5-x^4+x^2-2)$
$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5-\left(-x^4+x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
Intermediate steps
3
Simplify the product $-(-x^4+x^2-2)$
$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5+x^4-\left(x^2-2\right)\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
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Intermediate steps
4
Simplify the product $-(x^2-2)$
$\frac{\frac{d}{dx}\left(x^5-x^4+x^2-2\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
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5
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2+1\right)}{\left(x^2+1\right)^2}$
6
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(-2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)}{\left(x^2+1\right)^2}$
7
The derivative of the constant function ($-2$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\left(\frac{d}{dx}\left(x^2\right)+\frac{d}{dx}\left(1\right)\right)}{\left(x^2+1\right)^2}$
8
The derivative of the constant function ($1$) is equal to zero
$\frac{\left(\frac{d}{dx}\left(x^5\right)+\frac{d}{dx}\left(-x^4\right)+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2\right)}{\left(x^2+1\right)^2}$
9
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
$\frac{\left(\frac{d}{dx}\left(x^5\right)-\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(x^2\right)\right)\left(x^2+1\right)+\left(-x^5+x^4-x^2+2\right)\frac{d}{dx}\left(x^2\right)}{\left(x^2+1\right)^2}$
Intermediate steps
10
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{\left(5x^{4}-4x^{3}+2x\right)\left(x^2+1\right)+2\left(-x^5+x^4-x^2+2\right)x}{\left(x^2+1\right)^2}$
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Intermediate steps
11
Simplify the derivative
$\frac{6x+3x^{6}+5x^{4}-2x^{5}-4x^{3}}{\left(x^2+1\right)^2}$
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Final answer to the problem
$\frac{6x+3x^{6}+5x^{4}-2x^{5}-4x^{3}}{\left(x^2+1\right)^2}$