Final Answer
$\frac{6}{3x+2}+\frac{-4x^{3}}{x^4+7}$
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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
The logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator
$\frac{d}{dx}\left(\ln\left(\left(3x+2\right)^2\right)-\ln\left(x^4+7\right)\right)$
2
Using the power rule of logarithms: $\log_a(x^n)=n\cdot\log_a(x)$
$\frac{d}{dx}\left(2\ln\left(3x+2\right)-\ln\left(x^4+7\right)\right)$
3
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$\frac{d}{dx}\left(2\ln\left(3x+2\right)\right)+\frac{d}{dx}\left(-\ln\left(x^4+7\right)\right)$
Intermediate steps
4
The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function
$2\frac{d}{dx}\left(\ln\left(3x+2\right)\right)-\frac{d}{dx}\left(\ln\left(x^4+7\right)\right)$
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Intermediate steps
5
The derivative of the natural logarithm of a function is equal to the derivative of the function divided by that function. If $f(x)=ln\:a$ (where $a$ is a function of $x$), then $\displaystyle f'(x)=\frac{a'}{a}$
$2\left(\frac{1}{3x+2}\right)\frac{d}{dx}\left(3x+2\right)-\left(\frac{1}{x^4+7}\right)\frac{d}{dx}\left(x^4+7\right)$
Explain this step further
6
Multiplying the fraction by $-1$
$2\left(\frac{1}{3x+2}\right)\frac{d}{dx}\left(3x+2\right)+\frac{-1}{x^4+7}\frac{d}{dx}\left(x^4+7\right)$
7
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$2\left(\frac{1}{3x+2}\right)\left(\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(2\right)\right)+\frac{-1}{x^4+7}\frac{d}{dx}\left(x^4+7\right)$
8
The derivative of a sum of two or more functions is the sum of the derivatives of each function
$2\left(\frac{1}{3x+2}\right)\left(\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(2\right)\right)+\frac{-1}{x^4+7}\left(\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(7\right)\right)$
9
The derivative of the constant function ($2$) is equal to zero
$2\left(\frac{1}{3x+2}\right)\frac{d}{dx}\left(3x\right)+\frac{-1}{x^4+7}\left(\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(7\right)\right)$
10
The derivative of the constant function ($7$) is equal to zero
$2\left(\frac{1}{3x+2}\right)\frac{d}{dx}\left(3x\right)+\frac{-1}{x^4+7}\frac{d}{dx}\left(x^4\right)$
Intermediate steps
11
The derivative of the linear function times a constant, is equal to the constant
$6\left(\frac{1}{3x+2}\right)\frac{d}{dx}\left(x\right)+\frac{-1}{x^4+7}\frac{d}{dx}\left(x^4\right)$
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Intermediate steps
12
The derivative of the linear function is equal to $1$
$6\left(\frac{1}{3x+2}\right)+\frac{-1}{x^4+7}\frac{d}{dx}\left(x^4\right)$
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13
Multiply the fraction and term
$\frac{6}{3x+2}+\frac{-1}{x^4+7}\frac{d}{dx}\left(x^4\right)$
Intermediate steps
14
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
$\frac{6}{3x+2}+4\left(\frac{-1}{x^4+7}\right)x^{3}$
Explain this step further
15
Multiplying the fraction by $4x^{3}$
$\frac{6}{3x+2}+\frac{-4x^{3}}{x^4+7}$
Final Answer$\frac{6}{3x+2}+\frac{-4x^{3}}{x^4+7}$