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Math formulas: Higher-order Derivatives

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Definitions and properties

Second derivative

f=ddx(dydx)d2ydx2 f'' = \frac{d}{dx} \left(\frac{dy}{dx}\right) - \frac{d^2y}{dx^2}

Higher-Order derivative

f(n)=(f(n1)) f^{(n)} = \left( f^{(n-1)} \right)'
(f±g)(n)=f(n)± g(n) \left(f \, \pm \, g\right)^{(n)} = f^{(n)} \pm ~g^{(n)}

Leibniz's Formulas

(fg)=fg+2fg+fg(f \cdot g)'' = f'' \cdot g + 2 \cdot f'\cdot g' + f \cdot g''
(fg)=fg+3fg+3fg+fg(f \cdot g)''' = f''' \cdot g + 3 \cdot f''\cdot g' + 3 \cdot f'\cdot g'' + f \cdot g'''
(fg)(n)=f(n)g+nf(n1)g+n(n1)12f(n2)g++fg(n)(f \cdot g)^{(n)} = f^{(n)} \cdot g + n \cdot f^{(n-1)}\cdot g' + \frac{n(n-1)}{1\cdot2} \cdot f^{(n-2)} \cdot g'' + \dots + f \cdot g^{(n)}

Important Formulas

(xm)(n)=m!(mn)!xmn \left(x^m \right)^{(n)} = \frac{ m! }{(m-n)!} x^{m-n}
(xn)(n)=n! \left( x^n \right)^{(n)} = n!
(logax)(n)=(1)(n1)(n1)!xnlna \left( \log_a x \right)^{(n)} = \frac{(-1)^{(n-1)} \cdot (n-1)!}{x^n \cdot \ln a}
(lnn)(n)=(1)n1(n1)!xn (\ln n)^{(n)} = \frac{(-1)^{n-1}(n-1)!}{x^n}
(ax)(n)=axlnna \left( a^x \right)^{(n)} = a^x \cdot \ln^n a
(ex)(n)=ex \left( e^x \right)^{(n)} = e^x
(amx)(n)=mnamxlnna \left( a^{m \, x} \right)^{(n)} = m^n \, a^{m \cdot x} \ln^n a
(sinx)(n)=sin(x+nπ2) (\sin x)^{(n)} = \sin\left(x + \frac{n\,\pi}{2} \right)
(cosx)(n)=cos(x+nπ2) (\cos x)^{(n)} = \cos\left(x + \frac{n\,\pi}{2} \right)

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