find the derivatives in two ways : 1. by evaluating the integral and differentiating the result, 2. by differentiating the integral directly

Question

d/dx [definite integral b=sinx a=0 3t^2dt]

Answer 1

1. d/dx [ ∫ _{(from 0 to sinx)} 3t²dt ] = d/dx [ t³_{(from 0 to sinx)} ] = d/dx [ sin³x - sin³0 ] = 3sin²x cosx
2. d/dx [ ∫_{(from 0 to sinx)} 3t²dt ] = 3sin²x (sinx)' = 3sin²x cosx