No AI is used
  1. Etusivu
  2. Laskurit
  3. Laskurit: Differentiaali- ja integraalilaskenta I
  4. Differentiaali- ja integraalilaskin

Funktion $$$\sin{\left(x^{2} \right)}$$$ toinen derivaatta

Laskin laskee funktion $$$\sin{\left(x^{2} \right)}$$$ toisen derivaatan ja näyttää vaiheet.

Aiheeseen liittyvät laskurit: Derivointilaskin, Logaritmisen derivoinnin laskin

Jätä tyhjäksi automaattista tunnistusta varten.
Jätä tyhjäksi, jos et tarvitse derivaattaa tietyssä pisteessä.

Jos laskin ei laskenut jotakin tai olet havainnut virheen tai sinulla on ehdotus tai palaute, ole hyvä ja ota meihin yhteyttä.

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 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ käyttäen $$$n = 1$$$, toisin sanoen, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$- 2 x \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = - 2 x \sin{\left(x^{2} \right)} \frac{d}{dx} \left(x^{2}\right) + 2 \cos{\left(x^{2} \right)} {\color{red}\left(1\right)}$$

Sovella potenssisääntöä $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$, kun $$$n = 2$$$:

$$- 2 x \sin{\left(x^{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + 2 \cos{\left(x^{2} \right)} = - 2 x \sin{\left(x^{2} \right)} {\color{red}\left(2 x\right)} + 2 \cos{\left(x^{2} \right)}$$

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