If your loss is L=L1+L2 then the gradients at the input will sum.  Consider:
L1=f1(x)
L2=f2(x)
L=f1(x)+f2(x)
The derivative w.r.t x will be the derivative through f1 + the derivative through f2. This is because the derivative of a sum is the sum of the derivatives. If your combined loss is:
L=f1(x)*f2(x)
Then apply the chain rule and  you will obtain something similar. And so on.
Hope it helps!