Differentiating an accumulation function

The function $f(t) = \sin(t^2)$ is continuous everywhere, so the Fundamental Theorem of Calculus Part 1 applies directly to $F(x) = \int_0^x f(t)\, dt$.

FTC Part 1 states that $F'(x) = f(x)$, where $f$ is the integrand with $t$ replaced by $x$. Substituting $f(t) = \sin(t^2)$ gives $F'(x) = \sin(x^2)$.

The answer is the Fundamental Theorem of Calculus, Part 1. It states that if f is continuous, the derivative of the accumulation function F of x equals f of x itself. This is exactly why sine of x squared appeared immediately without any extra integration or chain rule work, since the upper limit was simply x.