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\( \newcommand{\dif}{\,\textrm{d}} \newcommand{\abs}[1]{\left\lvert #1 \right\rvert} \newcommand{\norm}[1]{\left\lVert #1 \right\rVert} \)

Differentiation

A list of common formulas, not in any way complete.
\[ \begin{align} (c)' &= 0 & (x)' &= 1 \\ (x^n)' &= n \cdot x^{n-1} & \left(\sqrt{x}\right)' &= \frac{1}{2\sqrt{x}} \\ (e^x)' &= e^x & (a^x)' &= a^x \cdot \ln a \\ (\ln x)' &= \frac{1}{x} & (\log_a x)' &= \frac{1}{x \cdot \ln a} \end{align} \]
\[ \begin{align} (f + g)'(x) &= f'(x) + g'(x) & (c \cdot f)'(x) &= c \cdot f'(x) \end{align} \] \[ \begin{align} (f \cdot g)'(x) &= f'(x) \cdot g(x) + f(x) \cdot g'(x) \\ (f_1 \cdot f_2 \cdot \ldots \cdot f_n)'(x) &= f'_1(x) \cdot f_2(x) \cdot \ldots \cdot f_n(x) \\ &+ f_1(x) \cdot f'_2(x) \cdot \ldots \cdot f_n(x) \\ &\dots \\ &+ f_1(x) \cdot f_2(x) \cdot \ldots \cdot f'_n(x) \end{align} \] \[ \begin{align} \left( \frac{1}{f} \right)'(x) &= \frac{-f'(x)}{f(x)^2} \\ \left( \frac{f}{g} \right)'(x) &= \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2} \\ \end{align} \] \[ \begin{equation} (g \circ f)'(x) = g'(f(x)) \cdot f'(x) \tag{chain rule} \end{equation} \]
\[ \begin{align} (\sin x)' &= \cos x & (\cos x)' &= -\sin x \\ (\tan x)' &= \frac{1}{\cos^2 x} & (\cot x)' &= \frac{-1}{\sin^2 x} \\ (\arcsin x)' &= \frac{1}{\sqrt{1 - x^2}} & (\arccos x)' &= \frac{-1}{\sqrt{1 - x^2}} \\ (\arctan x)' &= \frac{1}{1 + x^2} & (\operatorname{arccot} x)' &= \frac{-1}{1 + x^2} \end{align} \]