Benutzer:Arbol01/Ableitung 3x2 6x 12

Dies ist eine Arbeitsversion!

Funktion: $y=3x^{2}+6x+12$

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Methode1:
$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}{\frac {y-y_{0}}{x-x_{0}}}$

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}{\frac {(3x^{2}+6x+12)-(3x_{0}^{2}+6x_{0}+12)}{x-x_{0}}}$

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}{\frac {(3x^{2}+6x)-(3x_{0}^{2}+6x_{0})}{x-x_{0}}}$

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}{\frac {(3x^{2}-3x_{0}^{2})+(6x-6x_{0})}{x-x_{0}}}$

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}{\frac {3(x^{2}-x_{0}^{2})+6(x-x_{0})}{x-x_{0}}}$

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}{\frac {3(x+x_{0})(x-x_{0})+6(x-x_{0})}{x-x_{0}}}$

$x-x_{0}$ wird herausgekürzt

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}{\frac {3(x+x_{0})+6}{1}}$

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}3(x+x_{0})+6$

$y'={\frac {dy}{dx}}=\lim _{x\rightarrow x_{0}}6x_{0}+6$

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Methode2:
${\frac {d}{dx}}f(x)=\lim _{h\rightarrow 0}{\frac {f(x+h)-f(x)}{h}}=\lim _{h\rightarrow 0}{\frac {f(x)-f(x-h)}{h}}$



${\frac {d}{dx}}f(x)=\lim _{h\rightarrow 0}{\frac {(3(x+h)^{2}+6(x+h)+12)-(3x^{2}+6x+12)}{h}}$

${\frac {d}{dx}}f(x)=\lim _{h\rightarrow 0}{\frac {(3(x^{2}+2hx+h^{2})+6(x+h)+12)-(3x^{2}+6x+12)}{h}}$

${\frac {d}{dx}}f(x)=\lim _{h\rightarrow 0}{\frac {3x^{2}+6hx+3h^{2}+6x+6h+12-3x^{2}-6x-12}{h}}$

${\frac {d}{dx}}f(x)=\lim _{h\rightarrow 0}{\frac {6hx+3h^{2}+6h}{h}}$

Division durch h

${\frac {d}{dx}}f(x)=\lim _{h\rightarrow 0}{\frac {6x+3h+6}{1}}$

${\frac {d}{dx}}f(x)=\lim _{h\rightarrow 0}6x+6$

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Methode 2b:
$y=3x^{2}+6x+12$

$y+\delta y=3(x+\delta x)^{2}+6(x+\delta x)+12$

$\delta y=(3(x+\delta x)^{2}+6(x+\delta x)+12)-(3x^{2}+6x+12)$

$\delta y=(3(x^{2}+2x\delta x+\delta x^{2})+6(x+\delta x)+12)-(3x^{2}+6x+12)$

$\delta y=3x^{2}+6x\delta x+3\delta x^{2}+6x+6\delta x+12-3x^{2}-6x-12$

$\delta y=6x\delta x+3\delta x^{2}+6\delta x$

${\frac {\delta y}{\delta x}}=6x+3\delta x+6$

$y'={\frac {dy}{dx}}=\lim _{\delta x\rightarrow 0}{\frac {\delta y}{\delta x}}=6x+3\delta x+6$

$y'={\frac {dy}{dx}}=6x+6$

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Methode3:
$y=x^{2}$

${\sqrt {y}}=x$

$y'{\frac {1}{2{\sqrt {y}}}}=1$

$y'1=2{\sqrt {y}}$

$y'=2x$

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Methode4:
$y=x*x$

$y'=1*x+x*1$

$y'=2x$