Why is the Integral

$\displaystyle \frac{d}{dx}\int_1^{sinx} tant dt$

equal to

$\displaystyle tan(sinx) \cdot cosx$

Why don't you have to take the the antiderivative of tan(t). Like you would if it was

$\displaystyle \frac{d}{dx}\int_1^{sinx} t+5$

Also why is the lower limit not used when solving the problem?