$$$e^{x} \sin{\left(x \right)}$$$의 도함수

$$$f{\left(x \right)} = e^{x}$$$와 $$$g{\left(x \right)} = \sin{\left(x \right)}$$$에 대해 곱의 미분법칙 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$을 적용하십시오:

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

지수함수의 도함수는 $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$:

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

사인 함수의 도함수는 $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

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

간단히 하시오:

$$e^{x} \sin{\left(x \right)} + e^{x} \cos{\left(x \right)} = \sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}$$

따라서, $$$\frac{d}{dx} \left(e^{x} \sin{\left(x \right)}\right) = \sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}$$$.