# Is impulse response always differentiation of unit step response of a system?

I was trying to solve a question in which the transfer function of a system was asked, its unit step response being given: c(t) = 1-10exp(-t)

The method that the book followed was to first find out C(s) i.e. Laplace Transform(c(t)) = (1-9s)/(s(s+1)) Then find out the Laplace Transform of input i.e. R(s) = 1/s (since the input was step input) and finally the transfer function = C(s)/R(s) = (1-9s)/(s+1) (Answer).

But I tried to find out the transfer function by first calculating the impulse response of the system, which is equal to the time domain differetiation of unit step response. so, impulse response = d/dt(1-10exp(-t)) = (dirac delta(t))+10exp(-t) now transfer function will be Laplace Transform of Impulse response, So Transfer function = 1+(10/(s+1))

I can't figure out where is the mistake, why the answers differ when we apply a different approach.

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## migrated from mathematica.stackexchange.comSep 26 '13 at 20:56

This question came from our site for users of Mathematica.

Also asked on electronics.SE where it has already received a (correct but as yet unaccepted) answer. –  Dilip Sarwate Sep 27 '13 at 13:06