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Andra derivatan av $$$\tanh{\left(x \right)}$$$
Relaterade kalkylatorer: Derivata-beräknare, Kalkylator för logaritmisk derivering
Din inmatning
Bestäm $$$\frac{d^{2}}{dx^{2}} \left(\tanh{\left(x \right)}\right)$$$.
Lösning
Bestäm den första derivatan $$$\frac{d}{dx} \left(\tanh{\left(x \right)}\right)$$$
Derivatan av den hyperboliska tangenten är $$$\frac{d}{dx} \left(\tanh{\left(x \right)}\right) = \operatorname{sech}^{2}{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\tanh{\left(x \right)}\right)\right)} = {\color{red}\left(\operatorname{sech}^{2}{\left(x \right)}\right)}$$Alltså, $$$\frac{d}{dx} \left(\tanh{\left(x \right)}\right) = \operatorname{sech}^{2}{\left(x \right)}$$$.
Därefter, $$$\frac{d^{2}}{dx^{2}} \left(\tanh{\left(x \right)}\right) = \frac{d}{dx} \left(\operatorname{sech}^{2}{\left(x \right)}\right)$$$
Funktionen $$$\operatorname{sech}^{2}{\left(x \right)}$$$ är sammansättningen $$$f{\left(g{\left(x \right)} \right)}$$$ av två funktioner $$$f{\left(u \right)} = u^{2}$$$ och $$$g{\left(x \right)} = \operatorname{sech}{\left(x \right)}$$$.
Tillämpa kedjeregeln $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\operatorname{sech}^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\operatorname{sech}{\left(x \right)}\right)\right)}$$Tillämpa potensregeln $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$ med $$$n = 2$$$:
$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\operatorname{sech}{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\operatorname{sech}{\left(x \right)}\right)$$Återgå till den ursprungliga variabeln:
$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\operatorname{sech}{\left(x \right)}\right) = 2 {\color{red}\left(\operatorname{sech}{\left(x \right)}\right)} \frac{d}{dx} \left(\operatorname{sech}{\left(x \right)}\right)$$Derivatan av den hyperboliska sekansen är $$$\frac{d}{dx} \left(\operatorname{sech}{\left(x \right)}\right) = - \tanh{\left(x \right)} \operatorname{sech}{\left(x \right)}$$$:
$$2 \operatorname{sech}{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\operatorname{sech}{\left(x \right)}\right)\right)} = 2 \operatorname{sech}{\left(x \right)} {\color{red}\left(- \tanh{\left(x \right)} \operatorname{sech}{\left(x \right)}\right)}$$Alltså, $$$\frac{d}{dx} \left(\operatorname{sech}^{2}{\left(x \right)}\right) = - 2 \tanh{\left(x \right)} \operatorname{sech}^{2}{\left(x \right)}.$$$
Således, $$$\frac{d^{2}}{dx^{2}} \left(\tanh{\left(x \right)}\right) = - 2 \tanh{\left(x \right)} \operatorname{sech}^{2}{\left(x \right)}$$$.
Svar
$$$\frac{d^{2}}{dx^{2}} \left(\tanh{\left(x \right)}\right) = - 2 \tanh{\left(x \right)} \operatorname{sech}^{2}{\left(x \right)}$$$A