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