A type of differentiation under the integral sign

This question arises in a problem I'm working on in climate economics. Let $X\left( s \right)% MathType!MTEF!2!1!+-% feaahCart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x% e9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKk% Fr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam% iwamaabmaabaGaam4CaaGaayjkaiaawMcaaaaa!3B70!$be a real variable. How does one differentiate $\int\limits_0^t {X\left( s \right) ds} % MathType!MTEF!2!1!+-% feaahCart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x% e9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKk% Fr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8% qCaeaacaWGybWaaeWaaeaacaWGZbaacaGLOaGaayzkaaacciGae8hi% aaIaamizaiaadohaaSqaaiaaicdaaeaacaWG0baaniabgUIiYdaaaa!423E!$with respect to $X\left( t \right)% MathType!MTEF!2!1!+-% feaahCart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x% e9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKk% Fr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam% iwamaabmaabaGaamiDaaGaayjkaiaawMcaaaaa!3B71!$, the value of $X\left( \bullet \right)% MathType!MTEF!2!1!+-% feaahCart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x% e9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKk% Fr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam% iwamaabmaabaGaeyOiGClacaGLOaGaayzkaaaaaa!3BFD!$at the upper limit?I posed this to ChatGPT and it used the Dirac delta function to conclude that the answer is 1. However, I've found it sometimes makes math mistakes, and in any case I didn't understand its logic.

Any advice will be appreciated.