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Math Formulas: Higher-order Derivatives

Definitions and properties

Second derivative

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

Higher-Order derivative

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

Leibniz's Formulas

$$(f \cdot g)'' = f'' \cdot g + 2 \cdot f'\cdot g' + f \cdot g''$$
$$(f \cdot g)''' = f''' \cdot g + 3 \cdot f''\cdot g' + 3 \cdot f'\cdot g'' + f \cdot g'''$$
$$(f \cdot g)^{(n)} = f^{(n)} \cdot g + n \cdot f^{(n-1)}\cdot g' + \frac{n(n-1)}{1\cdot2} \cdot f^{(n-2)} \cdot g'' + \dots + f \cdot g^{(n)}$$

Important Formulas

$$ \left(x^m \right)^{(n)} = \frac{ m! }{(m-n)!} x^{m-n} $$
$$ \left( x^n \right)^{(n)} = n! $$
$$ \left( \log_a x \right)^{(n)} = \frac{(-1)^{(n-1)} \cdot (n-1)!}{x^n \cdot \ln a} $$
$$ (\ln n)^{(n)} = \frac{(-1)^{n-1}(n-1)!}{x^n} $$
$$ \left( a^x \right)^{(n)} = a^x \cdot \ln^n a $$
$$ \left( e^x \right)^{(n)} = e^x $$
$$ \left( a^{m \, x} \right)^{(n)} = m^n \, a^{m \cdot x} \ln^n a $$
$$ (\sin x)^{(n)} = \sin\left(x + \frac{n\,\pi}{2} \right) $$
$$ (\cos x)^{(n)} = \cos\left(x + \frac{n\,\pi}{2} \right) $$

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