<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] Note: This is an extensive edit to Shai Covo's answer.

For part ( C ) it is the following:
Pulling out a factor of $2$ from the denominator and the constant multiple of $3$ from the numerator leads to: $$ \frac{{3s - 15}}{{2s^2 - 4s + 10}} = \frac{3}{2}\frac{{s - 5}}{{s^2 - 2s + 5}} $$
Next we do Completing the Square for the polynomial $(s^2-2s+5)~$ in the denominator and re-writing the numerator constant $-5$ as $(-1-4)$ giving us:
$$ = \frac{3}{2} \cdot \frac{{s - 1 - 4}}{{(s - 1)^2 + 4}} $$
Let denominator equal $K$.
Separating the fraction into parts with them being $\dfrac{3}{2}\cdot\dfrac{(s-1)}{K}$ and $-3\cdot\dfrac{2}{K}$; after distributing the $\dfrac{3}{2}$ to the second term $-4$. This leads to the expression below now.
$$ =\frac{3}{2}\cdot\frac{{(s - 1)}}{{(s - 1)^2 + 2^2}} - 3\cdot\frac{2}{{(s - 1)^2 + 2^2 }}. $$ Now notice that $a = 1$ and $b = 2$. To see this, take a look at this table of Laplace Transforms for the form $e^{at} \cos(b t)$ and $e^{at} \sin (b t)$ located here $\longmapsto$ Table of $\mathcal Laplace~Transforms$ .
From the table of transforms, the inverse transform of $\dfrac{s-a }{(s-a )^2 + b ^2 }~$ and $~\dfrac{b}{{(s-a)^2 + b^2 }}$
are $e^{at} \cos(bt)$ and $e^{at} \sin (bt)$, respectively.
Hence the inverse laplace transform of our original problem is, $$ \mathcal{L^{-1}} \left\{ \dfrac{{3s - 15}}{{2s^2 - 4s + 10}} \right\} = \frac{3}{2}e^{t} \cos(2t) - 3e^{t} \sin(2t). $$
Hope this helps out.
Good~Luck
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