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Afgeleide van $$$e^{- \frac{1}{u^{2}}}$$$
Gerelateerde rekenmachines: Rekenmachine voor logaritmisch differentiëren, Rekenmachine voor impliciete differentiatie met stappen
Uw invoer
Bepaal $$$\frac{d}{du} \left(e^{- \frac{1}{u^{2}}}\right)$$$.
Oplossing
De functie $$$e^{- \frac{1}{u^{2}}}$$$ is de samenstelling $$$f{\left(g{\left(u \right)} \right)}$$$ van twee functies $$$f{\left(v \right)} = e^{v}$$$ en $$$g{\left(u \right)} = - \frac{1}{u^{2}}$$$.
Pas de kettingregel $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$ toe:
$${\color{red}\left(\frac{d}{du} \left(e^{- \frac{1}{u^{2}}}\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(e^{v}\right) \frac{d}{du} \left(- \frac{1}{u^{2}}\right)\right)}$$De afgeleide van de exponentiële functie is $$$\frac{d}{dv} \left(e^{v}\right) = e^{v}$$$:
$${\color{red}\left(\frac{d}{dv} \left(e^{v}\right)\right)} \frac{d}{du} \left(- \frac{1}{u^{2}}\right) = {\color{red}\left(e^{v}\right)} \frac{d}{du} \left(- \frac{1}{u^{2}}\right)$$Keer terug naar de oorspronkelijke variabele:
$$e^{{\color{red}\left(v\right)}} \frac{d}{du} \left(- \frac{1}{u^{2}}\right) = e^{{\color{red}\left(- \frac{1}{u^{2}}\right)}} \frac{d}{du} \left(- \frac{1}{u^{2}}\right)$$Pas de regel van de constante factor $$$\frac{d}{du} \left(c f{\left(u \right)}\right) = c \frac{d}{du} \left(f{\left(u \right)}\right)$$$ toe met $$$c = -1$$$ en $$$f{\left(u \right)} = \frac{1}{u^{2}}$$$:
$$e^{- \frac{1}{u^{2}}} {\color{red}\left(\frac{d}{du} \left(- \frac{1}{u^{2}}\right)\right)} = e^{- \frac{1}{u^{2}}} {\color{red}\left(- \frac{d}{du} \left(\frac{1}{u^{2}}\right)\right)}$$Pas de machtsregel $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ toe met $$$n = -2$$$:
$$- e^{- \frac{1}{u^{2}}} {\color{red}\left(\frac{d}{du} \left(\frac{1}{u^{2}}\right)\right)} = - e^{- \frac{1}{u^{2}}} {\color{red}\left(- \frac{2}{u^{3}}\right)}$$Dus, $$$\frac{d}{du} \left(e^{- \frac{1}{u^{2}}}\right) = \frac{2 e^{- \frac{1}{u^{2}}}}{u^{3}}$$$.
Antwoord
$$$\frac{d}{du} \left(e^{- \frac{1}{u^{2}}}\right) = \frac{2 e^{- \frac{1}{u^{2}}}}{u^{3}}$$$A