Question

why do we need to put negative sign in the "kt" (example dx/dt= -kt) in the application of differential equations in the decomposition of radioactive substances?

Answer 1

because its decreasing,,and not increasing
for eg. if you say rate is inc by 2,,its different in notation than decreasing by 2
dx/dt =2 <--inc
3x/dt=-2 <-- dec

Answer 2

because it is stupid
\(k\) is a variable and so of course can be positive or negative