Formelsamling/Matematik/Derivering och integrering

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Deriveringsregler

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

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

Derivator

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

Kedjeregeln

Om y = f(x) och z = g(x) är två deriverbara funktioner, gäller

${\frac {dy}{dx}}={\frac {dy}{dz}}\cdot {\frac {dz}{dx}}\,\!$

För den sammansatta funktionen y = f(g(x)) gäller

$y'=f'(g(x))\cdot g'(x)\,\!$

Gränsvärden

$\lim _{x\rightarrow \infty }\left(1+{\frac {1}{x}}\right)^{x}=e\qquad \lim _{x\rightarrow 0}{\frac {e^{x}-1}{x}}=1$

$\lim _{x\rightarrow 0}{\frac {\sin x}{x}}=1\quad {\text{(x uttryckt i radianer)}}$

$\lim _{x\rightarrow 0^{+}}{\frac {\ln(x+1)}{x}}=1$

Integraler

En godtycklig primitiv funktion till f(x) kan skrivas

$\int f(x)\ dx=F(x)+C\,\!$ (C är en konstant)

Integralberäkning och räkneregler

$\int _{a}^{b}f(x)\ dx=F(b)-F(a)=\left[F(x)\right]_{a}^{b}\,\!$

$\int _{a}^{b}f(x)\ dx=\int _{a}^{c}f(x)\ dx+\int _{c}^{b}f(x)\ dx\,\!$

$\int _{a}^{b}[f(x)+g(x)]\ dx=\int _{a}^{b}f(x)\ dx+\int _{a}^{b}g(x)\ dx\,\!$

$\int _{a}^{b}k\cdot f(x)\ dx=\ k\cdot \int _{a}^{b}f(x)\ dx\,\!$

Partiell integration och substitution

$\int f(x)\ g(x)\ dx=\ F(x)\ g(x)\ -\int F(x)\ g'(x)\ dx\,\!$

$\int _{a}^{b}f(g(x))\ g'(x)\ dx=\int _{g(a)}^{g(b)}f(t)\ dt\,\!$