Real Analysis - Basic Differentiation Rules

$C' = 0$, where $C$ is a constant

$(C \cdot x)' = C$, where $C$ is a constant

$(x^n)' = n \cdot x^{n-1}$, $x \in \mathbb{R}$, $n \in \mathbb{Z}$

$(x^a)' = a \cdot x^{a-1}$, $x \in \mathbb{R}$, $x \gt 0$, $a \in \mathbb{R}$

$(\sin{x})' = \cos{x}$

$(\cos{x})' = -\sin{x}$

$(\tan{x})' = (tg\ {x})' = 1 / \cos^2{x}$

$(\cot{x})' = (cotg\ {x})' = -1 / \sin^2{x}$

$(\arcsin{x})' = 1 / \sqrt{1-x^2}$

$(\arccos{x})' = -1 / \sqrt{1-x^2}$

$(arctan\ {x})' = (arctg\ {x})' = 1 / (1 + x^2)$

$(arccot\ {x})' = (arccotg\ {x})' = -1 / (1 + x^2)$

$(e^x)' = e^x$

$(a^x)' = a^x \cdot \ln{a}\ \ \ $ ($a \gt 0$ is a constant)

$(\ln{x})' = 1 / x$

$(\log_{a}x)' = 1 / (x \cdot \ln{a})\ \ \ $ ($a \gt 0$ is a constant)

$$(u + v)' = u' + v'$$

$$(u - v)' = u' - v'$$

$$(uv)' = u' \cdot v + v' \cdot u $$

$$\left(\frac{u}{v}\right)' = \frac{u'v - v'u}{v^2}$$

$$(f(g(x)))' = f'(g(x)) \cdot g'(x)$$