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Funktion $$$\sin{\left(x^{2} \right)}$$$ toinen derivaatta
Aiheeseen liittyvät laskurit: Derivointilaskin, Logaritmisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right)$$$.
Ratkaisu
Laske ensimmäinen derivaatta $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)$$$
Funktio $$$\sin{\left(x^{2} \right)}$$$ on kahden funktion $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ ja $$$g{\left(x \right)} = x^{2}$$$ yhdistelmä $$$f{\left(g{\left(x \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$Sinin derivaatta on $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right)$$Palaa alkuperäiseen muuttujaan:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) = \cos{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right)$$Sovella potenssisääntöä $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$, kun $$$n = 2$$$:
$$\cos{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = \cos{\left(x^{2} \right)} {\color{red}\left(2 x\right)}$$Näin ollen, $$$\frac{d}{dx} \left(\sin{\left(x^{2} \right)}\right) = 2 x \cos{\left(x^{2} \right)}$$$.
Seuraavaksi $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = \frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right)$$$
Sovella vakion kerroinsääntöä $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ käyttäen $$$c = 2$$$ ja $$$f{\left(x \right)} = x \cos{\left(x^{2} \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right)\right)} = {\color{red}\left(2 \frac{d}{dx} \left(x \cos{\left(x^{2} \right)}\right)\right)}$$Sovella tulon derivointisääntöä $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ funktioille $$$f{\left(x \right)} = x$$$ ja $$$g{\left(x \right)} = \cos{\left(x^{2} \right)}$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x^{2} \right)}\right)\right)} = 2 {\color{red}\left(\frac{d}{dx} \left(x\right) \cos{\left(x^{2} \right)} + x \frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right)\right)}$$Funktio $$$\cos{\left(x^{2} \right)}$$$ on kahden funktion $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ ja $$$g{\left(x \right)} = x^{2}$$$ yhdistelmä $$$f{\left(g{\left(x \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$2 x {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x^{2} \right)}\right)\right)} + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = 2 x {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(x^{2}\right)\right)} + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$Kosinin derivaatta on $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$$2 x {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = 2 x {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$Palaa alkuperäiseen muuttujaan:
$$- 2 x \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = - 2 x \sin{\left({\color{red}\left(x^{2}\right)} \right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$Sovella potenssisääntöä $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$, kun $$$n = 2$$$:
$$- 2 x \sin{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right) = - 2 x \sin{\left(x^{2} \right)} {\color{red}\left(2 x\right)} + 2 \cos{\left(x^{2} \right)} \frac{d}{dx} \left(x\right)$$Sovella potenssisääntöä $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ käyttäen $$$n = 1$$$, toisin sanoen, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)} {\color{red}\left(1\right)}$$Näin ollen, $$$\frac{d}{dx} \left(2 x \cos{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$.
Siispä $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$.
Vastaus
$$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x^{2} \right)}\right) = - 4 x^{2} \sin{\left(x^{2} \right)} + 2 \cos{\left(x^{2} \right)}$$$A