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