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