We can derive the McLaurin series for #sinh(x)# from the one othe exponential function: as for every #n#:

#[(d^n)/(dx^n) e^x ]_(x=0) = e^0=1#

the Mc Laurin series for #e^x# is:

#e^x=sum_(n=0)^oo x^n/(n!)#

Now as:

#sinhx = (e^x-e^(-x))/2#

We have:

#sinhx = 1/2[sum_(n=0)^oo x^n/(n!)-sum_(n=0)^oo (-x)^n/(n!)]#

and it is easy to see that for #n# even the terms are the same and just cancel each other, so that just the odd order terms remain:

#sinhx = 1/2[sum_(k=0)^oo x^(2k+1)/((2k+1)!)-sum_(k=0)^oo (-1)^(2k+1)x^(2k+1)/((2k+1)!)] = 1/2[sum_(k=0)^oo x^(2k+1)/((2k+1)!)+sum_(k=0)^oo x^(2k+1)/((2k+1)!)] = sum_(k=0)^oo x^(2k+1)/((2k+1)!)#

We can reach the same conclusion directly, noting that:

#d/(dx) sinhx = coshx#

#d^2/(dx^2) sinhx = d/(dx)coshx = sinhx#

so that all derivatives of odd order equal #coshx# and all derivatives of even order equal #sinhx#

But #sinh(0) = 0# and #cosh(0) = 1# yielding the same result.