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Funktion $$$e^{- x} \sin{\left(x \right)}$$$ derivaatta pisteessä $$$x = c$$$
Aiheeseen liittyvät laskurit: Logaritmisen derivoinnin laskin, Vaiheittainen implisiittisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)$$$ ja laske sen arvo kohdassa $$$x = c$$$.
Ratkaisu
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 $$$f{\left(x \right)} = e^{- x}$$$ ja $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(e^{- x}\right) \sin{\left(x \right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$Funktio $$$e^{- x}$$$ on kahden funktion $$$f{\left(u \right)} = e^{u}$$$ ja $$$g{\left(x \right)} = - x$$$ yhdistelmä $$$f{\left(g{\left(x \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$$\sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(e^{- x}\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(- x\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Eksponenttifunktion derivaatta on $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$$\sin{\left(x \right)} {\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)} {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Palaa alkuperäiseen muuttujaan:
$$e^{{\color{red}\left(u\right)}} \sin{\left(x \right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = e^{{\color{red}\left(- x\right)}} \sin{\left(x \right)} \frac{d}{dx} \left(- x\right) + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Sovella vakion kerroinsääntöä $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ käyttäen $$$c = -1$$$ ja $$$f{\left(x \right)} = x$$$:
$$e^{- x} \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(- x\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) = e^{- x} \sin{\left(x \right)} {\color{red}\left(- \frac{d}{dx} \left(x\right)\right)} + e^{- x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Sinin derivaatta on $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$- e^{- x} \sin{\left(x \right)} \frac{d}{dx} \left(x\right) + e^{- x} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = - e^{- x} \sin{\left(x \right)} \frac{d}{dx} \left(x\right) + e^{- x} {\color{red}\left(\cos{\left(x \right)}\right)}$$Sovella potenssisääntöä $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ käyttäen $$$n = 1$$$, toisin sanoen, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- e^{- x} \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + e^{- x} \cos{\left(x \right)} = - e^{- x} \sin{\left(x \right)} {\color{red}\left(1\right)} + e^{- x} \cos{\left(x \right)}$$Sievennä:
$$- e^{- x} \sin{\left(x \right)} + e^{- x} \cos{\left(x \right)} = \sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}$$Näin ollen, $$$\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right) = \sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}$$$.
Lopuksi laske derivaatta pisteessä $$$x = c$$$.
$$$\left(\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)\right)|_{\left(x = c\right)} = \sqrt{2} e^{- c} \cos{\left(c + \frac{\pi}{4} \right)}$$$
Vastaus
$$$\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right) = \sqrt{2} e^{- x} \cos{\left(x + \frac{\pi}{4} \right)}$$$A
$$$\left(\frac{d}{dx} \left(e^{- x} \sin{\left(x \right)}\right)\right)|_{\left(x = c\right)} = \sqrt{2} e^{- c} \cos{\left(c + \frac{\pi}{4} \right)}\approx 1.414213562373095 e^{- c} \cos{\left(c + \frac{\pi}{4} \right)}$$$A