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

In the given question, system is assumed to be causal, input is unit step ,therefore step response will be $[1-10e^{-t}]u(t)$. Now , let us apply both the methods :

1: $c(t) =[ 1-10e^{-t}]u(t)$

$H(s)= \frac{C(s)}{X(s)} = s\left[\frac1s-\frac{10}{s+1}\right] = \frac{1-9s}{s+1}$

2: $c(t) = [1-10e^{-t}]u(t)$ \begin{align*} h(t) &= \delta[c(t)]/dt \\ &= \delta[u(t)]/dt -10 \delta[e^{-t}u(t)]/dt) &\text{ UV form differentiation} \\ &= \delta(t) + 10e^{-t}u(t) - 10e^{-t}\delta(t) \\ &= \delta(t)-10\delta(t)+ 10e^{-t}u(t) & x(t)d(t) = x(0)d(t) \\ &= 10e^{-t}u(t) - 9\delta(t) \\ \end{align*}

$H(s) = \frac{10}{s+1} - 9 = \frac{1-9s}{s+1}$