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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 $\displaystyle f'(x)=\frac{a'}{a}$
Divide fractions $\frac{1}{\frac{\left(3x+2\right)^2}{x^4+7}}$ with Keep, Change, Flip: $a\div \frac{b}{c}=\frac{a}{1}\div\frac{b}{c}=\frac{a}{1}\times\frac{c}{b}=\frac{a\cdot c}{b}$
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 ${\displaystyle h'(x) = \frac{f'(x) \cdot g(x) - g'(x) \cdot f(x)}{g(x)^2}}$
Multiplying fractions $\frac{x^4+7}{\left(3x+2\right)^2} \times \frac{\frac{d}{dx}\left(\left(3x+2\right)^2\right)\left(x^4+7\right)-\left(3x+2\right)^2\frac{d}{dx}\left(x^4+7\right)}{\left(x^4+7\right)^2}$
Simplify the fraction by $x^4+7$
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
The derivative of a sum of two or more functions is the sum of the derivatives of each function
The derivative of a sum of two or more functions is the sum of the derivatives of each function
Simplify the product $-(\frac{d}{dx}\left(x^4\right)+\frac{d}{dx}\left(7\right))$
The derivative of the constant function ($2$) is equal to zero
The derivative of the constant function ($7$) is equal to zero
Multiply $-1$ times $0$
The derivative of the linear function times a constant, is equal to the constant
The derivative of the linear function is equal to $1$
The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$
Simplify the derivative