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