Standard Proof of Fundamental Theorem of Calculus as (Backward) Magics

Here is a copy of the standard proof of the Fundamental Theorem of Calculus from the standard test book Calculus: A Complete Course, by Adams and Essex, which represents Backward Magics and not Forward Reason:

We see that Adams starts with the primitive function $F(x)=\int_a^xf(t)\, dt$ and proves that it satisfies the differential equation $F^{\,\prime} (x) = f(x)$, which is Backward Magic.