Matemaatika:Funktsiooni tuletis

Diferentseerimine

Valemid

${\begin{matrix}x'=1\\(x^{2})'=2x\\(x^{\alpha })'=\alpha x^{\alpha -1}\\({\sqrt {x}})'={\frac {1}{2{\sqrt {x}}}}\\({\frac {1}{x}})'=-{\frac {1}{x^{2}}}\\(e^{x})'=e^{x}\\(ln(x))'={\frac {1}{x}}\\(log_{a}x)'={\frac {1}{xln(a)}}\end{matrix}}$${\begin{matrix}(a)'=0\\(a*u)'=a*(u)'\\(u+v)'=u'+v'\\(u-v)'=u'-v'\\(u*v)'=u'*v+u*v'\\({\frac {u}{v}})'={\frac {u'*v-u*v'}{v^{2}}}\\(a^{x})'=a^{x}ln(a)\end{matrix}}$${\begin{matrix}(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}}}\\(arccottan(x))'=-{\frac {1}{1+x^{2}}}\end{matrix}}$

Meeldetuletuseks

Tähemärk a on konstant, mis on kindla ühe arvväärtusega element. Arv e=~2,71828182845905...
${\begin{matrix}ln(a*b)=ln(a)+ln(b)\\ln({\frac {a}{b}})=ln(a)-ln(b)\\ln(a^{b})=b*ln(a)\end{matrix}}$

$e=\lim _{x\to \infty }(1+{\frac {1}{x}})^{x}$

Logaritmiline diferentseerimine

• korrutan mõlemad pooled naturaallogaritmiga läbi
• diferentseerin x järgi
• avaldan tuletise
• asendan y tema algväärtusega

Ülesanne 1

${\begin{matrix}y=x^{x}\end{matrix}}$,leida y' ${\begin{matrix}y=x^{x}|ln\\ln(y)=x*ln(x)|{\frac {d}{dx}}\\{\frac {1}{y}}*y'=1*ln(x)+x*{\frac {1}{x}}\\y'=y(ln(x)+1)=x^{x}(ln(x)+1)\end{matrix}}$

Ülesanne 2

${\begin{matrix}y=x^{x^{x}}\end{matrix}}$,leida y' ${\begin{matrix}y=x^{x^{x}}|ln\\ln(y)=x^{x}*ln(x)|ln\\ln(ln(y))=x*ln(x)+ln(ln(x))|{\frac {d}{dx}}\\{\frac {1}{ln(y)}}*{\frac {1}{y}}*y'=1*ln(x)+x*{\frac {1}{x}}+{\frac {1}{ln(x)}}*{\frac {1}{x}}\\y'=y*ln(y)*(ln(x)+1+{\frac {1}{x*ln(x)}})\\y'=x^{x^{x}}*x^{x}*ln(x)*(ln(x)+1+{\frac {1}{x*ln(x)}})\end{matrix}}$