Surjective isometries on function spaces with derivatives

Let $A$ be a complex Banach space with a norm $\mathrm{\parallel }f\mathrm{\parallel }=\mathrm{\parallel }f{\mathrm{\parallel }}_X+\mathrm{\parallel }d(f){\mathrm{\parallel }}_Y$ for $f\in A$, where $d$ is a complex linear map from $A$ onto a Banach space $B$, and $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_K$ represents the supremum norm on a compact Hausdorff space $K$. In this paper, we characterize surjective isometries on $(A,\mathrm{\parallel }\cdot \mathrm{\parallel })$, which may be nonlinear. This unifies former results on surjective isometries between specific function spaces.