$($b$)$ Suppose the following systems take $x(t)$ as the input and $y(t)$ as the

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corresponding output. Find the impulse response $h(t).$

$y(t)=x(t-7)$;

II$)y(t)={\displaystyle \underset{-}{\overset{t}{}}}x(-7)d.$

$($c$)$ Consider the LTI system shown as below:

Express the system impulse response as a function of the impulse responses of the subsystems.

$($d$)$ Suppose the systems with impulse response $h(t)$ take $x(t)$ as the input.

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Find the output $y(t).$

$x(t)=u(1-t)$ and $h(t)=e^{-t}u(t-2)$

$($e$)$ For the convolution between two time$-$domain signals $f(t)$ and $g(t),$ the

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differentiation property is:

$\frac{d}{dt}(f(t)**g(t))=\frac{df(t)}{dt}**g(t)=f(t)**\frac{dg(t)}{dt}$

Calculate $\frac{d}{dt}(e^{-t}**u(t)).$