differentiation and intergration of sinx/x and d/dt P(t)=P`(t)

(1) problem

differentiation [intergration of sint/t], limit from 0 to x^2

let say f(t)=t, F(t)=t^2/2, if f(t)=sint/t, F(t)=intergration of sint/t

d/dx [intergration of sint/t] limit from 0 to x^2

d/dx [F(t)] limit from 0 to x^2

{d/dx [F(t)} = differentiation of F(t)

then {d/dx [F(t)} = [f(t)] limit from 0 to x^2

f(x^2)-f(0)

(sin x^2/x)-(sin0/0)

(sin x^2/x)-1

in this correct?(Wondering)

(2) problem

d/dt P(t)=P`(t)

t=0, P`(0)=0

t->infinite, then P`(infinite)=f(h)/h

in this right?(Wondering)