$$$\cos^{2}{\left(x \right)}$$$의 도함수

함수 $$$\cos^{2}{\left(x \right)}$$$는 두 함수 $$$f{\left(u \right)} = u^{2}$$$와 $$$g{\left(x \right)} = \cos{\left(x \right)}$$$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:

$${\color{red}\left(\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{2}\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$

거듭제곱법칙 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$을 $$$n = 2$$$에 적용합니다:

$${\color{red}\left(\frac{d}{du} \left(u^{2}\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = {\color{red}\left(2 u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$

역치환:

$$2 {\color{red}\left(u\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) = 2 {\color{red}\left(\cos{\left(x \right)}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$

코사인의 도함수는 $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$입니다:

$$2 \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = 2 \cos{\left(x \right)} {\color{red}\left(- \sin{\left(x \right)}\right)}$$

간단히 하시오:

$$- 2 \sin{\left(x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)}$$

따라서, $$$\frac{d}{dx} \left(\cos^{2}{\left(x \right)}\right) = - \sin{\left(2 x \right)}$$$.