## 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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