Answer
$$
F^{\prime}(x) = \frac{d}{dx} \left[ x \sec ^{-1}\left(x^{3}\right) \right] =\frac{3}{\sqrt{x^{6}-1}}+\sec ^{-1}\left(x^{3}\right)
$$
Work Step by Step
$$
F(x)=x \sec ^{-1}\left(x^{3}\right)
$$
Differentiating both sides of this equation, using multiply Rule we have
$$
\begin{aligned}
\frac{d}{dx} \left[ F(x) \right] &= \frac{d}{dx} \left[ x \sec ^{-1}\left(x^{3}\right) \right] \\
F^{\prime}(x) &=x \cdot \frac{1}{x^{3} \sqrt{\left(x^{3}\right)^{2}-1}} \frac{d}{d x}\left(x^{3}\right)+\sec ^{-1}\left(x^{3}\right) \cdot 1 \\
&=\frac{x\left(3 x^{2}\right)}{x^{3} \sqrt{x^{6}-1}}+\sec ^{-1}\left(x^{3}\right) \\
&=\frac{3}{\sqrt{x^{6}-1}}+\sec ^{-1}\left(x^{3}\right)
\end{aligned}
$$
