Richard Feynman (Nobel Physicist) proved it in high school using a funny Calculus: “Differentiating under Integral” — is it legitimate to do so ? Of course it is ! by “The Fundamental Theorem of Calculus”
Note: We were thought in high school the “HOW” of calculating (such as integration and differentiation), but not the “WHY” (the Theories behind). Richard Feynman was unique in exploring the WHY since high school, it helped later he was assigned by President Reagan to investigate the 1986 ‘Challenger’ disaster ?