How do you differentiate f(x)=3ln(e^(x^2+1)/(2x^3)) ?

$f ' \left(x\right) = 3 \cdot \frac{\left(2 {x}^{2} - 3\right)}{x}$

Explanation:

The following formulas may be use in this problem

$\frac{d}{\mathrm{dx}} \left(\frac{u}{v}\right) = \frac{v \cdot \frac{d}{\mathrm{dx}} \left(u\right) - u \cdot \frac{d}{\mathrm{dx}} \left(v\right)}{v} ^ 2$

also

$\frac{d}{\mathrm{dx}} \left(\ln u\right) = \frac{1}{u} \cdot \frac{d}{\mathrm{dx}} \left(u\right)$

From the given $f \left(x\right) = 3 \ln \left({e}^{{x}^{2} + 1} / \left(2 {x}^{3}\right)\right)$

$f ' \left(x\right) = \frac{d}{\mathrm{dx}} \left(3 \ln \left({e}^{{x}^{2} + 1} / \left(2 {x}^{3}\right)\right)\right)$

$f ' \left(x\right) = 3 \cdot \frac{d}{\mathrm{dx}} \left(\ln \left({e}^{{x}^{2} + 1} / \left(2 {x}^{3}\right)\right)\right)$

$f ' \left(x\right) = 3 \cdot \left(\frac{1}{{e}^{{x}^{2} + 1} / \left(2 {x}^{3}\right)}\right) \cdot \frac{d}{\mathrm{dx}} \left({e}^{{x}^{2} + 1} / \left(2 {x}^{3}\right)\right)$

$f ' \left(x\right) = 3 \cdot \left(\frac{2 {x}^{3}}{{e}^{{x}^{2} + 1}}\right) \cdot \left(\frac{\left(2 {x}^{3}\right) \frac{d}{\mathrm{dx}} \left({e}^{{x}^{2} + 1}\right) - \left({e}^{{x}^{2} + 1}\right) \frac{d}{\mathrm{dx}} \left(2 {x}^{3}\right)}{2 {x}^{3}} ^ 2\right)$

$f ' \left(x\right) = 3 \cdot \left(\frac{2 {x}^{3}}{{e}^{{x}^{2} + 1}}\right) \cdot \frac{\left(2 {x}^{3}\right) \left({e}^{{x}^{2} + 1}\right) \frac{d}{\mathrm{dx}} \left({x}^{2} + 1\right) - \left({e}^{{x}^{2} + 1}\right) \left(6 {x}^{2}\right)}{2 {x}^{3}} ^ 2$

$f ' \left(x\right) = 3 \cdot \left(\frac{2 {x}^{3}}{{e}^{{x}^{2} + 1}}\right) \cdot \frac{\left(2 {x}^{3}\right) \left({e}^{{x}^{2} + 1}\right) \left(2 x\right) - \left({e}^{{x}^{2} + 1}\right) \left(6 {x}^{2}\right)}{2 {x}^{3}} ^ 2$

$f ' \left(x\right) = 3 \cdot \left(\frac{2 {x}^{3}}{\cancel{{e}^{{x}^{2} + 1}}}\right) \cdot \frac{\left(4 {x}^{4} - 6 {x}^{2}\right) \cancel{{e}^{{x}^{2} + 1}}}{2 {x}^{3}} ^ 2$

$f ' \left(x\right) = 3 \cdot \frac{\left(2 {x}^{4} - 3 {x}^{2}\right)}{{x}^{3}}$

$f ' \left(x\right) = 3 \cdot \frac{\left(2 {x}^{2} - 3\right)}{x}$

God bless...I hope the explanation is useful.