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1 Deriveringsregler
2 Derivator
3 Integraler
- 3.1 Integralberäkning och räkneregler
- 3.2 Partiell integration och substitution

[redigera] Deriveringsregler

Låt $x\in \mathbb{R}$ , $n\in \mathbb{Z}^+$ och $a\in \mathbb{R}$ .

Funktion	Derivata	Primitiv funktion
$y=f\left ( x \right )$	$f^\prime(x)\,\!$	$\int f(x)dx\,\!$
$x^a\,\!$	$ax^{a-1}\,\!$	$\frac{x^{a+1}}{a+1}$
$\frac {1}{x} (x\ne 0) \,\!$	$- \frac {1}{x^2}\,\!$	$\ln \|x\|\,\!$
$\sqrt x\,\!$	$\frac {1}{2 \sqrt x}\,\!$	$\frac{2}{3}x^{3/2}$
$\frac{1}{x^n}$	$-\frac{n}{x^{n+1}}$	$\begin{cases} \frac{x^{1-n}}{1-n}, & n\ne 1 \\ \ln\|x\|, & n=1 \end{cases}$
$e^{x}\,\!$	$e^{x}\,\!$	$e^{x}\,\!$
$a^x \quad (a>0,\ a \ne 1)$	$a^x\cdot\ln a$	$\frac{a^x}{\ln a}$
$\ln x\,\!$	$\frac {1}{x} \quad x\ne 0 \,\!$	$x\ln {x-x}\,\!$
$\log _a x \quad (a,x>0,\ a\ne 1)$	$\frac{1}{x \ln a}$	$\frac{x\ln{x-x}}{\ln a}$
$\frac {1}{x+a}\,\!$	$\frac{-1}{(x+a)^2}$	$\ln \|x+a\|\,\!$
$\frac {1}{(x-a)(x-b)}\,\!$		$\frac {1}{a-b}\ln \frac {\|x-a\|}{\|x-b\|}\,\!$
$\frac {1}{(x-a)^2}\,\!$		$-\frac {1}{x-a}\,\!$
$\frac {1}{x^2+a^2}\,\!$		$\frac {1}{a} \arctan \frac {x}{a}\,\!$
$\frac {x}{x^2+a^2}\,\!$		$\frac {1}{2} \ln (x^2+a^2)\,\!$
$\sqrt {x^2+a^2}\,\!$		$\frac {x}{2} \sqrt {x^2+a^2} \ + \frac {a^2}{2} \ln (x+ \sqrt {x^2+a^2})\,\!$
$\sqrt {x^2-a^2}\,\!$		$\frac {x}{2} \sqrt {x^2-a^2} \ - \frac {a^2}{2} \ln \|x+ \sqrt {x^2-a^2}\|\,\!$
$\sqrt {a^2-x^2}\,\!$		$\frac {x}{2} \sqrt {a^2-x^2} \ + \frac {a^2}{2} \ arc \ \sin \frac {x}{a}\,\!$
$\frac {1}{\sqrt {x^2+a^2}}\,\!$		$\ln (x+ \sqrt {x^2+a^2})\,\!$
$\frac {1}{\sqrt {x^2-a^2}}\,\!$		$\ln \|x+ \sqrt {x^2-a^2}\|\,\!$
$\frac {1}{\sqrt {a^2-x^2}}\,\!$		$\arcsin \frac{x}{a}\,\!$
$\sin x\,\!$	$\cos x\,\!$	$-\cos x\,\!$
$\sin^2 x\,\!$		$\frac {1}{2} (x- \sin x \cos x)\,\!$
$\frac {1}{\sin x}=\csc x\,\!$	$-\cos x \csc^2 x\,\!$	$\ln\|\tan \frac {x}{2}\|\,\!$
$\frac {1}{\sin^2 x}\,\!$		$- \cot x\,\!$
$\cos x\,\!$	$-\sin x\,\!$	$\sin x\,\!$
$\cos^2 x\,\!$		$\frac {1}{2} (x+ \sin x \cos x)\,\!$
$\frac {1}{\cos x}=\sec x\,\!$	$\sin x \sec^2 x\,\!$	$\ln \|\tan\ (\frac {x}{2} + \frac {\pi}{4})\|\,\!$
$\frac {1}{\cos^2 x}\,\!$		$\tan x\,\!$
$\tan x\,\!$	$1+ \tan^2 x=\frac {1}{\cos^2 x}\,\!$	$-\ln \|\cos x\|\,\!$
$\cot x\,\!$	$-(1+ \cot^2 x)=- \frac {1}{\sin^2 x}\,\!$	$\ln \|\sin x\|\,\!$

[redigera] Derivator

[redigera] Derivata av summa, produkt och kvot

f(x) och g(x) är deriverbara funktioner

Funktion		Derivata
$h(x)=f(x)+g(x)\,\!$		$h'(x)=f'(x)+g'(x)\,\!$
$h(x)=f(x) \cdot g(x)\,\!$		$h'(x)=f(x) \cdot g'(x)+ g(x) \cdot f'(x)\,\!$
	eller	$\frac {h'(x)}{h(x)}= \frac {f'(x)}{f(x)}+ \frac {g'(x)}{g(x)} \quad (f(x)\ och\ g(x) \ne 0)\,\!$
$h(x)= \frac {f(x)}{g(x)} \quad (g(x) \ne 0)\,\!$		$h'(x)= \frac {g(x) \cdot f'(x)-f(x) \cdot g'(x)}{[g(x)]^2}\,\!$
	eller	$\frac {h'(x)}{h(x)}= \frac {f'(x)}{f(x)}- \frac {g'(x)}{g(x)} \quad (f(x)\ och\ g(x) \ne 0)\,\!$