Math formulas and cheat sheets for Higher-order Derivatives

Higher-order Derivatives

Definitions and properties

Second derivative

f′′=ddx(dydx)d2ydx2

Higher-Order derivative

f(n)=(f(n1))
(f±g)(n)=f(n)± g(n)

Leibniz's Formulas

(fg)′′=f′′g+2fg+fg′′
(fg)′′′=f′′′g+3f′′g+3fg′′+fg′′′
(fg)(n)=f(n)g+nf(n1)g+n(n1)12f(n2)g′′++fg(n)

Important Formulas

(xm)(n)=m!(mn)!xmn
(xn)(n)=n!
(logax)(n)=(1)(n1)(n1)!xnlna
(lnn)(n)=(1)n1(n1)!xn
(ax)(n)=axlnna
(ex)(n)=ex
(amx)(n)=mnamxlnna
(sinx)(n)=sin(x+nπ2)
(cosx)(n)=cos(x+nπ2)