Final answer to the problem
$\frac{5}{\left(2x+1\right)^{2}}$
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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 Find the derivative using logarithmic differentiation Find the derivative using the definition Suggest another method or feature
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1
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{d}{dx}\left(\frac{x}{4x^2+2x}\right)+\frac{d}{dx}\left(\frac{-3}{2x+1}\right)$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{\frac{d}{dx}\left(x\right)\left(4x^2+2x\right)-x\frac{d}{dx}\left(4x^2+2x\right)}{\left(4x^2+2x\right)^2}+\frac{d}{dx}\left(\frac{-3}{2x+1}\right)$
3
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
$\frac{\frac{d}{dx}\left(x\right)\left(4x^2+2x\right)-x\frac{d}{dx}\left(4x^2+2x\right)}{\left(4x^2+2x\right)^2}+\frac{\frac{d}{dx}\left(-3\right)\left(2x+1\right)+3\frac{d}{dx}\left(2x+1\right)}{\left(2x+1\right)^2}$
4
The derivative of the constant function ($-3$) is equal to zero
$\frac{\frac{d}{dx}\left(x\right)\left(4x^2+2x\right)-x\frac{d}{dx}\left(4x^2+2x\right)}{\left(4x^2+2x\right)^2}+\frac{0\left(2x+1\right)+3\frac{d}{dx}\left(2x+1\right)}{\left(2x+1\right)^2}$
5
Any expression multiplied by $0$ is equal to $0$
$\frac{\frac{d}{dx}\left(x\right)\left(4x^2+2x\right)-x\frac{d}{dx}\left(4x^2+2x\right)}{\left(4x^2+2x\right)^2}+\frac{0+3\frac{d}{dx}\left(2x+1\right)}{\left(2x+1\right)^2}$
6
$x+0=x$, where $x$ is any expression
$\frac{\frac{d}{dx}\left(x\right)\left(4x^2+2x\right)-x\frac{d}{dx}\left(4x^2+2x\right)}{\left(4x^2+2x\right)^2}+\frac{3\frac{d}{dx}\left(2x+1\right)}{\left(2x+1\right)^2}$
Intermediate steps
7
The derivative of the linear function is equal to $1$
$\frac{4x^2+2x-x\frac{d}{dx}\left(4x^2+2x\right)}{\left(4x^2+2x\right)^2}+\frac{3\frac{d}{dx}\left(2x+1\right)}{\left(2x+1\right)^2}$
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8
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{4x^2+2x-x\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(4x^2+2x\right)^2}+\frac{3\frac{d}{dx}\left(2x+1\right)}{\left(2x+1\right)^2}$
9
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{4x^2+2x-x\left(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(2x\right)\right)}{\left(4x^2+2x\right)^2}+\frac{3\left(\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(1\right)\right)}{\left(2x+1\right)^2}$
10
Simplify the product $-(\frac{d}{dx}\left(4x^2\right)+\frac{d}{dx}\left(2x\right))$
$\frac{4x^2+2x+\left(-\frac{d}{dx}\left(4x^2\right)-\frac{d}{dx}\left(2x\right)\right)x}{\left(4x^2+2x\right)^2}+\frac{3\left(\frac{d}{dx}\left(2x\right)+\frac{d}{dx}\left(1\right)\right)}{\left(2x+1\right)^2}$
11
The derivative of the constant function ($1$) is equal to zero
$\frac{4x^2+2x+\left(-\frac{d}{dx}\left(4x^2\right)-\frac{d}{dx}\left(2x\right)\right)x}{\left(4x^2+2x\right)^2}+\frac{3\frac{d}{dx}\left(2x\right)}{\left(2x+1\right)^2}$
Intermediate steps
12
The derivative of the linear function times a constant, is equal to the constant
$\frac{4x^2+2x+\left(-\frac{d}{dx}\left(4x^2\right)-2\frac{d}{dx}\left(x\right)\right)x}{\left(4x^2+2x\right)^2}+\frac{3\frac{d}{dx}\left(2x\right)}{\left(2x+1\right)^2}$
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Intermediate steps
13
The derivative of the linear function times a constant, is equal to the constant
$\frac{4x^2+2x+\left(-\frac{d}{dx}\left(4x^2\right)-2\frac{d}{dx}\left(x\right)\right)x}{\left(4x^2+2x\right)^2}+\frac{6\frac{d}{dx}\left(x\right)}{\left(2x+1\right)^2}$
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Intermediate steps
14
The derivative of the linear function is equal to $1$
$\frac{4x^2+2x+\left(-\frac{d}{dx}\left(4x^2\right)-2\right)x}{\left(4x^2+2x\right)^2}+\frac{6\frac{d}{dx}\left(x\right)}{\left(2x+1\right)^2}$
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Intermediate steps
15
The derivative of the linear function is equal to $1$
$\frac{4x^2+2x+\left(-\frac{d}{dx}\left(4x^2\right)-2\right)x}{\left(4x^2+2x\right)^2}+\frac{6}{\left(2x+1\right)^2}$
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Intermediate steps
16
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
$\frac{4x^2+2x+\left(-4\frac{d}{dx}\left(x^2\right)-2\right)x}{\left(4x^2+2x\right)^2}+\frac{6}{\left(2x+1\right)^2}$
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Intermediate steps
17
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{4x^2+2x+\left(-8x-2\right)x}{\left(4x^2+2x\right)^2}+\frac{6}{\left(2x+1\right)^2}$
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Intermediate steps
18
Simplify the derivative
$\frac{5}{\left(2x+1\right)^{2}}$
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Final answer to the problem
$\frac{5}{\left(2x+1\right)^{2}}$