Find the derivative of ln((((3x+2)^5)/(x^4+7))^0.5)
Answer
$\frac{\frac{1}{2}\left(\frac{15\left(2+3x\right)^{4}\left(7+x^4\right)-4\left(2+3x\right)^5x^{3}}{\left(7+x^4\right)^2}\right)\left(7+x^4\right)}{\left(2+3x\right)^{5}}$
Step-by-step explanation
Problem
$\frac{d}{dx}\left(\ln\left(\sqrt{\frac{\left(3x+2\right)^5}{x^4+7}}\right)\right)$
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 $\displaystyle f'(x)=\frac{a'}{a}$
$\frac{1}{\sqrt{\frac{\left(2+3x\right)^5}{7+x^4}}}\cdot\frac{d}{dx}\left(\sqrt{\frac{\left(2+3x\right)^5}{7+x^4}}\right)$
Answer
$\frac{\frac{1}{2}\left(\frac{15\left(2+3x\right)^{4}\left(7+x^4\right)-4\left(2+3x\right)^5x^{3}}{\left(7+x^4\right)^2}\right)\left(7+x^4\right)}{\left(2+3x\right)^{5}}$