1. **State the problem:** Find the derivative $y'$ if $y = \sec\left(e^{x^4}\right)$. 2. **Recall the formula:** The derivative of $\sec(u)$ with respect to $x$ is $\frac{d}{dx} \sec(u) = \sec(u) \tan(u) \frac{du}{dx}$. 3. **Identify the inner function:** Here, $u = e^{x^4}$. We need $\frac{du}{dx}$. 4. **Differentiate the inner function:** $$\frac{du}{dx} = \frac{d}{dx} e^{x^4} = e^{x^4} \cdot \frac{d}{dx} (x^4) = e^{x^4} \cdot 4x^3.$$ 5. **Apply the chain rule:** $$y' = \sec\left(e^{x^4}\right) \tan\left(e^{x^4}\right) \cdot 4x^3 e^{x^4}.$$ 6. **Final answer:** $$y' = 4x^3 e^{x^4} \sec\left(e^{x^4}\right) \tan\left(e^{x^4}\right).$$ This matches the fourth option given in the problem.