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Second derivatives

Each of the functions sinh and cosh is equal to its second derivative, that is: d 2 d x 2 sinh x = sinh x {\displaystyle {\frac {d^{2}}{dx^{2}}}\sinh x=\sinh x} d 2 d x 2 cosh x = cosh x. {\displaystyle {\frac {d^{2}}{dx^{2}}}\cosh x=\cosh x\,.}

All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x {\displaystyle e^{x}} and e − x {\displaystyle e^{-x}}.