f(x)=sinh(x)cos(x).
Let S=sinh(x) and C=cosh(x), so dS/dx=C, dC/dx=S.
Let s=sin(x) and c=cos(x), so ds/dx=c, dc/dx=-s.
sinh(0)=0, cosh(0)=1, sin(0)=0, cos(0)=1.
f(x)=Sc; f(0)=0
f⁽¹⁾(x)=Cc-Ss; f⁽¹⁾(0)=1
f⁽²⁾(x)=Sc-Cs-Cs-Sc=-2Cs; f⁽²⁾(0)=0
f⁽³⁾(x)=-2Ss-2Cc; f⁽³⁾(0)=-2
f⁽⁴⁾(x)=-2Cs-2Sc-2Sc+2Cs=-4Sc=-4f(x); f⁽⁴⁾(0)=0
f⁽⁵⁾(x)=-4f⁽¹⁾(x); f⁽⁵⁾(0)=-4
f⁽⁶⁾(x)=-4f⁽²⁾(x); f⁽⁶⁾(0)=0
f⁽⁷⁾(x)=-4f⁽³⁾(x); f⁽⁷⁾(0)=8
f⁽⁸⁾(x)=-4f⁽⁴⁾(x); f⁽⁸⁾(0)=0
f⁽⁹⁾(x)=-4f⁽⁵⁾(x)=16f⁽¹⁾(x); f⁽⁹⁾(0)=16
f⁽¹¹⁾(x)=-4f⁽⁷⁾(x)=16f⁽³⁾(x); f⁽¹¹⁾(0)=-32
The Maclaurin term is f⁽ⁿ⁾xⁿ/n!.
The series is x-(2x³/3!+4x⁵/5!)+(8x⁷/7!+16x⁹/9!)-(32x¹¹/11!+64x¹³/13!)...
This can be written:
x+∑[n ∈[0,∞)](-1)ⁿ⁺¹(2²ⁿ⁺¹x⁴ⁿ⁺³/(4n+3)!+2²ⁿ⁺²x⁴ⁿ⁺⁵/(4n+5)!).