<s>[INST] Laplace transform and Differentiation help I need to know how to differentiate something like these 
a) Differentiate with respect to s  $$ b/(s^2 +b^2)$$
b) Solve the given equation 
                               $$f(t) = s(1/s^2 + 2/s^3)$$
c) Reverse Laplace transform of $$(3s - 15)/(2s^2 -4s + 10)$$
 [/INST] For c),
$$
\frac{{3s - 15}}{{2s^2  - 4s + 10}} = \frac{3}{2}\frac{{s - 5}}{{s^2  - 2s + 5}} = \frac{3}{2}\frac{{s - 1 - 4}}{{(s - 1)^2  + 4}} = \frac{3}{2}\frac{{s - 1}}{{(s - 1)^2  + 2^2}} - 3\frac{2}{{(s - 1)^2  + 2^2 }}.
$$
Now let $\alpha = -1$ and $\omega = 2$, and consider 8-9 of this table.
EDIT: By 9 and 8 of that table, the inverse transforms of $\frac{{s + \alpha }}{{(s + \alpha )^2  + \omega ^2 }}$ and  $\frac{\omega}{{(s + \alpha)^2  + \omega^2 }}$ are $e^{-\alpha t} \cos(\omega t)$ and $e^{-\alpha t} \sin (\omega t)$, respectively. Hence the inverse transform of $\frac{{3s - 15}}{{2s^2  - 4s + 10}}$ is 
$$
\frac{3}{2}e^t \cos (2t) - 3e^t \sin (2t).
$$
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