Exponential decaying step response in LTI System
You're forgetting the discontinuity at $t=0$. Your step response is actually

$$Y(t)=(1+e^{-0.1t})u(t)\tag{1}$$

where $u(t)$ is the unit step function. Differentiating $(1)$ gives

$$h(t)=-0.1e^{-0.1t}u(t)+(1+e^{-0.1t})\delta(t)=-0.1e^{-0.1t}u(t)+2\delta(t)\tag{2}$$

where $\delta(t)$ is the Dirac delta impulse. Convolving $(2)$ with a unit step gives the original step response:

$$\begin{align}\int_{-\infty}^{\infty}h(\tau)u(t-\tau)d\tau&amp;=-0.1u(t)\int_{0}^{t}e^{-0.1\tau}d\tau+2\int_{-\infty}^{t}\delta(\tau)d\tau\\
&amp;=u(t)(e^{-0.1t}-1)+2u(t)\\&amp;=(1+e^{-0.1t})u(t)\end{align}$$