Higher-order Derivatives Formulas

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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