Derivatan av $$$e^{x} \sin{\left(x \right)}$$$
Relaterade kalkylatorer: Kalkylator för logaritmisk derivering, Räknare för implicit derivering med steg
Din inmatning
Bestäm $$$\frac{d}{dx} \left(e^{x} \sin{\left(x \right)}\right)$$$.
Lösning
Tillämpa produktregeln $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ med $$$f{\left(x \right)} = e^{x}$$$ och $$$g{\left(x \right)} = \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(e^{x} \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(e^{x}\right) \sin{\left(x \right)} + e^{x} \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$Derivatan av exponentialfunktionen är $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$:
$$e^{x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} = e^{x} \frac{d}{dx} \left(\sin{\left(x \right)}\right) + \sin{\left(x \right)} {\color{red}\left(e^{x}\right)}$$Derivatan av sinus är $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$e^{x} \sin{\left(x \right)} + e^{x} {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = e^{x} \sin{\left(x \right)} + e^{x} {\color{red}\left(\cos{\left(x \right)}\right)}$$Förenkla:
$$e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} = \sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}$$Alltså, $$$\frac{d}{dx} \left(e^{x} \sin{\left(x \right)}\right) = \sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}$$$.
Svar
$$$\frac{d}{dx} \left(e^{x} \sin{\left(x \right)}\right) = \sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}$$$A