Use chain, logarithmic, product and quotient differentiation

The derivative is $\dfrac{3e^{3x}}{1+(e^{3x})^2}=\dfrac{3e^{3x}}{1+e^{6x}}$.

$\dfrac{d}{dx}5^{2x}=5^{2x}\ln5\cdot2=(2\ln5)5^{2x}$.

Let $y=5^{2x}>0$. Taking logarithms gives $\ln y=2x\ln5$, so $y\prime/y=2\ln5$ and $y\prime=(2\ln5)5^{2x}$.

Write $ax(x^3+1)^{-1/2}$. Its derivative is $a(x^3+1)^{-1/2}-\frac32ax^3(x^3+1)^{-3/2}$, the printed form. Combining the terms gives $\dfrac{a(2-x^3)}{2(x^3+1)^{3/2}}$, for $x>-1$.

The quotient-rule numerator is $a\sqrt{x^3+1}-ax\cdot\frac12(x^3+1)^{-1/2}(3x^2)$ and its denominator is $x^3+1$. This simplifies to the same printed two-term form and hence the same combined fraction.