Machten en wortels
Regel
Voorbeeld
Voorwaarde
a^0=1
x^0=2^0=(x+3)^0=1
a\neq0
a^{1}=a
3^1=3
geen
a^{-n}=\frac{1}{a^n}
x^{-3}=\frac{1}{x^3}
a\neq0
a^{\frac{1}{2}}=\sqrt{a}
x^{1/2}=\sqrt{x}
a>0, n>0
a^{\frac{1}{n}}=\sqrt[n]{a}
x^{1/5}=\sqrt[5]{x}
a>0, n>0
a^p \cdot a^q=a^{p+q}
x^3 \cdot x^2=x^{3+2}=x^5
a>0
\frac{a^p}{a^q}=a^{p-q}
\frac{x^3}{x^2}=x^{3-2}=x^1=x
a>0
(ab)^n=a^nb^n
(3x)^3=3^3b^3=27b^3
a>0, b>0
(a^p)^q=a^{pq}
(x^3)^4=x^{3 \cdot 4}=x^{12}
a>0
\sqrt{A\cdot B}=\sqrt{A}\cdot\sqrt{B}
\sqrt{12}=\sqrt{4}\cdot\sqrt{3}=2\sqrt{3}
A\ge 0, B\ge 0
\sqrt{\frac{A}{B}}=\frac{\sqrt{A}}{\sqrt{B}}
\sqrt{\frac{16}{25}}=\frac{\sqrt{16}}{\sqrt{25}}=\frac{4}{5}
A\ge 0, B> 0
Logaritmen
Regel
Voorbeeld
Voorwaarde
^g\log(1)=0
^4\log(1)= \,^4\log(4^0)=0
g >0, g\neq 1
^g\log(g^p)=p
^4\log(16)= \,^4\log(4^2)=2
g >0, g\neq 1
^g\log{a} = \frac{^p\log(a)}{^p\log(g)}
^8\log(4) = \frac{^2\log(4)}{^2\log(8)}= \frac{^2\log(2^2)}{^2\log(2^3)}=\frac{2}{3}
g>0, a>0, p>0, p \neq 1
^g\log(a) +\, ^g\log(b)= \,^g\log(a\cdot b)
^3\log(4) +\, ^3\log(x)= \,^3\log(4x)
g >0, g\neq 1, a>0, b>0
^g\log(a) -\, ^g\log(b)= \,^g\log(\frac{a}{b})
^3\log(4) -\, ^3\log(x)= \,^3\log(\frac{4}{x})
g >0, g\neq 1, a>0, b>0
^g\log(a^p)= p \cdot\, ^g\log(a)
\ln(x^7)=7 \cdot \ln(x)
g >0, g\neq 1, a>0
Goniometrie
\sin(-a)=-\sin(a)
\cos(-a)=\cos(a)
\sin(\frac{1}{2}\pi -a)=\cos(a)
\cos(\frac{1}{2}\pi -a)=\sin(a)
\sin(\pi – a)=\sin(a)
\cos(\pi – a)=-\cos(a)
\sin ^2(a) + \cos ^2(a) = 1
\tan(a)=\frac{\sin(a)}{\cos (a)}
\sin(2a)=2\sin(a)\cos(a)
\cos(2a)=\cos ^2(a) – \sin ^2(a)=2\cos ^2(a)-1 = 1-2\sin ^2(a)
\sin(a+b)=\sin(a)\cos(b)+\cos(a)\sin(b)
\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)
\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b)
\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)
\sin(a)+\sin(b)=2\sin(\frac{a+b}{2})\cos(\frac{a-b}{2})
\sin(a)-\sin(b)=2\sin(\frac{a-b}{2})\cos(\frac{a+b}{2})
\cos(a)+\cos(b)=2\cos(\frac{a+b}{2})\cos(\frac{a-b}{2})
\cos(a)-\cos(b)=-2\sin(\frac{a+b}{2})\sin(\frac{a-b}{2})
Goniometrische vergelijkingen oplossen
Standaardvergelijking
Oplossing
\cos(A)=B
A=\cos^{-1}(B)+k\cdot2\pi of A=-\cos^{-1}(B)+k\cdot2\pi
\sin(A)=B
A=\sin^{-1}(B)+k\cdot2\pi of A=\pi-\sin^{-1}(B)+k\cdot2\pi
\tan(A)=B
A=\tan^{-1}(B)+k\cdot\pi
\sin(A)=\sin(B)
A=B+k\cdot 2\pi of A=\pi-B+k\cdot 2\pi
\cos(A)=\cos(B)
A=B+k\cdot 2\pi of A=-B+k\cdot 2\pi
Differentiëren (standaardregels)
Regel
Functie
Afgeleide
Constante maal f
f(x)=c\cdot g(x)
f'(x)=c\cdot g'(x)
Somregel
f(x)=g(x)+h(x)
f'(x)=g'(x)+h'(x)
Productregel
f(x)=g(x)\cdot h(x)
f'(x)=g(x)\cdot h'(x)+g'(x)\cdot h(x)
Quotiëntregel
f(x)=\frac{t(x)}{n(x)}
f'(x)=\frac{t'(x)\cdot n(x)-t(x)\cdot n'(x)}{(n(x))^2}
Kettingregel
f(g(x))
f'(g(x))\cdot g'(x)
Afgeleiden van standaardfuncties
Functie
Afgeleide
f(x)=c
f'(x)=0
f(x)=x^n
f'(x)=n\cdot x^{n-1}
f(x)=e^x
f'(x)=e^x
f(x)=g^x
f'(x)=g^x\cdot \ln(g)
f(x)=\ln(x)
f'(x)=\frac{1}{x}
f(x)=^g\log(x) met g>0, g\neq1
f'(x)=\frac{1}{x\cdot \ln(g)}
f(x)=\sin(x)
f'(x)=\cos(x)
f(x)=\cos(x)
f'(x)=-\sin(x)
f(x)=\tan(x)
f'(x)=\frac{1}{\cos ^2(x)}
Primitieven van standaardfuncties
Functie
Primitieve
f(x)=a
F(x)=a\cdot x+C
f(x)=x^n met n\neq-1
F(x)=\frac{1}{n+1}x^{n+1}+C
f(x)=\frac{1}{x}
F(x)=\ln|x|+C
f(x)=e^x
F(x)=e^x+C
f(x)=g^x
F(x)=\frac{1}{\ln(g)}g^x+C
f(x)=ln(x)
F(x)=x\cdot \ln(x)-x+C
f(x)=^g\log(x) met g>0, g\neq1
F(x)=\frac{1}{\ln(g)}\cdot (x\cdot \ln(x)-x)+C
f(x)=\sin(x)
F(x)=-\cos(x)+C
f(x)=\cos(x)
F(x)=\sin(x)+C
Integreren
Wat bereken je
Integraal
Oppervlak tussen f(x) en de x-as op interval x\in [a, b]
A=\int_{a}^{b} f(x)dx
Omwentelingslichaam van f(x) om de x-as op interval x\in [a, b]
V=\pi\cdot\int_{a}^{b}(f(x))^2dx
Omwentelingslichaam van f(x) om de y-as op interval y\in [y_1, y_2]
V=\pi\cdot\int_{y_1}^{y_2}(f^{-1}(y))^2dy
Lengte van f(x) op interval x\in [a, b]
L=\int_{a}^{b}\sqrt{1+(f'(x))^2}dx
Driehoeken
Regel
Formule
Stelling van Pythagoras (alleen voor rechthoekige driehoeken)
a^2+b^2=c^2
Sinusregel
\frac{a}{\sin(\alpha)}=\frac{b}{\sin(\beta)}=\frac{c}{\sin(\gamma)}
Cosinusregel
c^2=a^2+b^2-2ab\cdot\cos(\gamma)