That doesn't look correct to me. It looks like you make two mistakes that offset each other. When applying the derivative rule, we use the initial value at 0-, and the initial value of cos(t) u(t) is 0 at t=0-. This is exactly what I want you to see and understand. Then when you transform cos(t) δ(t) you try to integrate from 0- to 0+, which typically we dont' try to do. There may be a way to do this (although I've never tried) using special rules, but we don't even try because we can use the sifting property of the delta function to get the answer more directly. Whenever you integrate a function with the δ(t), you simple extract the value of that function at t=0. Remember that the 0- and 0+ indicate the instruction to include or exclude the weird behavior of certain functions at t=0. Hence, the integral of a any function with δ(t) from -∞ to +∞ is the same as from 0- to 0+ , both of which will include the impulse function. The answer just amounts to plugging t=0 into the function you are trying to integrate. Hence the Laplace transform of cos(t) δ(t) is simply cos(0) exp(0)=1.
So, you had the initial value of 1 and the transform of cos(t) δ(t) as 0, but it should be the initial value is 0 and the transform of cos(t) δ(t) is 1.
Remember what I said before that the initial value at t=0- is zero for most functions that we deal with, which includes all derivatives of δ(t) and you are correct that the n'th derivative of δ(t) transforms to s^n.
I'll have to scan through this later to see if I can spot the discrepancy.
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