$$$x$$$에 대한 $$$\cos{\left(b - x \right)}$$$의 도함수

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

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

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

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

역치환:

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

합/차의 도함수는 도함수들의 합/차이다:

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

상수의 도함수는 $$$0$$$입니다:

$$- \left({\color{red}\left(\frac{db}{dx}\right)} - \frac{d}{dx} \left(x\right)\right) \sin{\left(b - x \right)} = - \left({\color{red}\left(0\right)} - \frac{d}{dx} \left(x\right)\right) \sin{\left(b - x \right)}$$

멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$\sin{\left(b - x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = \sin{\left(b - x \right)} {\color{red}\left(1\right)}$$

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