+130466 See if this helps, Max : http://tutorial.math.lamar.edu/Classes/DE/Laplace_Table.aspx
Series expansion at x=0:
(script capital l)(0)+x (script capital l)'(0)+1/2 x^2 (script capital l)''(0)+1/6 x^3 ((script capital l)^(3)(0)+(script capital l)'(0))+1/24 x^4 ((script capital l)^(4)(0)+4 (script capital l)''(0))+O(x^5)
(Taylor series)
Possible derivation:
d/dx((script capital l)(sinh(x)))
Using the chain rule, d/dx((script capital l)(sinh(x))) = ( d(script capital l)(u))/( du) ( du)/( dx), where u = sinh(x) and ( d)/( du)((script capital l)(u)) = (script capital l)'(u):
= (d/dx(sinh(x))) (script capital l)'(sinh(x))
The derivative of sinh(x) is cosh(x):
Answer: | = cosh(x) (script capital l)'(sinh(x))
