= Solution
Apply the <Leibniz integral rule>, including both moving boundaries and the explicit parameter in the integrand. The upper bound has <derivative> $-\sin x$, the lower bound has <derivative> $\cos x$, and the integrand has <partial derivative> $-t^4e^{-xt^4}$ with respect to $x$. Therefore
$$
\boxed{f'(x)=-\sin x\,e^{-x\cos^4x}-\cos x\,e^{-x\sin^4x}-\int_{\sin x}^{\cos x}t^4e^{-xt^4}\,dt.}
$$
The formula remains valid when the bounds coincide or occur in reversed order, by the oriented <definite integral> convention. The integrand and its <partial derivative> are continuous on every finite integration interval.
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