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Andraderivatan av $$$b^{x}$$$ med avseende på $$$x$$$
Relaterade kalkylatorer: Derivata-beräknare, Kalkylator för logaritmisk derivering
Din inmatning
Bestäm $$$\frac{d^{2}}{dx^{2}} \left(b^{x}\right)$$$.
Lösning
Bestäm den första derivatan $$$\frac{d}{dx} \left(b^{x}\right)$$$
Tillämpa potenslagen $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ med $$$n = b$$$:
$${\color{red}\left(\frac{d}{dx} \left(b^{x}\right)\right)} = {\color{red}\left(b^{x} \ln\left(b\right)\right)}$$Alltså, $$$\frac{d}{dx} \left(b^{x}\right) = b^{x} \ln\left(b\right)$$$.
Därefter, $$$\frac{d^{2}}{dx^{2}} \left(b^{x}\right) = \frac{d}{dx} \left(b^{x} \ln\left(b\right)\right)$$$
Tillämpa konstantfaktorregeln $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ med $$$c = \ln\left(b\right)$$$ och $$$f{\left(x \right)} = b^{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(b^{x} \ln\left(b\right)\right)\right)} = {\color{red}\left(\ln\left(b\right) \frac{d}{dx} \left(b^{x}\right)\right)}$$Tillämpa potenslagen $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$ med $$$n = b$$$:
$$\ln\left(b\right) {\color{red}\left(\frac{d}{dx} \left(b^{x}\right)\right)} = \ln\left(b\right) {\color{red}\left(b^{x} \ln\left(b\right)\right)}$$Alltså, $$$\frac{d}{dx} \left(b^{x} \ln\left(b\right)\right) = b^{x} \ln^{2}\left(b\right)$$$.
Således, $$$\frac{d^{2}}{dx^{2}} \left(b^{x}\right) = b^{x} \ln^{2}\left(b\right)$$$.
Svar
$$$\frac{d^{2}}{dx^{2}} \left(b^{x}\right) = b^{x} \ln^{2}\left(b\right)$$$A