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$$$e^{\frac{x}{2}}$$$의 도함수
사용자 입력
$$$\frac{d}{dx} \left(e^{\frac{x}{2}}\right)$$$을(를) 구하시오.
풀이
함수 $$$e^{\frac{x}{2}}$$$는 두 함수 $$$f{\left(u \right)} = e^{u}$$$와 $$$g{\left(x \right)} = \frac{x}{2}$$$의 합성함수 $$$f{\left(g{\left(x \right)} \right)}$$$이다.
연쇄법칙 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$을(를) 적용하십시오:
$${\color{red}\left(\frac{d}{dx} \left(e^{\frac{x}{2}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(\frac{x}{2}\right)\right)}$$지수함수의 도함수는 $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(\frac{x}{2}\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(\frac{x}{2}\right)$$역치환:
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(\frac{x}{2}\right) = e^{{\color{red}\left(\frac{x}{2}\right)}} \frac{d}{dx} \left(\frac{x}{2}\right)$$상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = \frac{1}{2}$$$와 $$$f{\left(x \right)} = x$$$에 적용합니다:
$$e^{\frac{x}{2}} {\color{red}\left(\frac{d}{dx} \left(\frac{x}{2}\right)\right)} = e^{\frac{x}{2}} {\color{red}\left(\frac{\frac{d}{dx} \left(x\right)}{2}\right)}$$멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\frac{e^{\frac{x}{2}} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{2} = \frac{e^{\frac{x}{2}} {\color{red}\left(1\right)}}{2}$$따라서, $$$\frac{d}{dx} \left(e^{\frac{x}{2}}\right) = \frac{e^{\frac{x}{2}}}{2}$$$.
정답
$$$\frac{d}{dx} \left(e^{\frac{x}{2}}\right) = \frac{e^{\frac{x}{2}}}{2}$$$A