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\frac{d}{dx}\left(\ln\left(\frac{\left(3x+2\right)^2}{x^4+7}\right)\right)

Find the derivative of ln(((3x+2)^2)/(x^4+7))

Answer

$\frac{6\left(7+x^4\right)\left(2+3x\right)-4x^{3}\left(4+12x+9x^2\right)}{7\left(2+3x\right)^2+x^4\left(2+3x\right)^2}$

Step-by-step explanation

Problem

$\frac{d}{dx}\left(\ln\left(\frac{\left(3x+2\right)^2}{x^4+7}\right)\right)$
1

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 $\displaystyle f'(x)=\frac{a'}{a}$

$\frac{1}{\frac{\left(2+3x\right)^2}{7+x^4}}\cdot\frac{d}{dx}\left(\frac{\left(2+3x\right)^2}{7+x^4}\right)$

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Answer

$\frac{6\left(7+x^4\right)\left(2+3x\right)-4x^{3}\left(4+12x+9x^2\right)}{7\left(2+3x\right)^2+x^4\left(2+3x\right)^2}$

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$\frac{d}{dx}\left(\ln\left(\frac{\left(3x+2\right)^2}{x^4+7}\right)\right)$

Main topic:

Differential calculus

Used formulas:

5. See formulas

Time to solve it:

0.57 seconds