$$$e^{x} \sin{\left(x \right)}$$$の導関数

積の微分法 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を $$$f{\left(x \right)} = e^{x}$$$ と $$$g{\left(x \right)} = \sin{\left(x \right)}$$$ に適用する:

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

指数関数の微分は$$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$です:

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

正弦関数の導関数は$$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:

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

簡単化せよ:

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

したがって、$$$\frac{d}{dx} \left(e^{x} \sin{\left(x \right)}\right) = \sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}$$$。