$$$z$$$에 대한 $$$e^{x} + \sin{\left(y z \right)}$$$의 도함수

합/차의 도함수는 도함수들의 합/차이다:

$${\color{red}\left(\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right)\right)} = {\color{red}\left(\frac{d}{dz} \left(e^{x}\right) + \frac{d}{dz} \left(\sin{\left(y z \right)}\right)\right)}$$

상수의 도함수는 $$$0$$$입니다:

$${\color{red}\left(\frac{d}{dz} \left(e^{x}\right)\right)} + \frac{d}{dz} \left(\sin{\left(y z \right)}\right) = {\color{red}\left(0\right)} + \frac{d}{dz} \left(\sin{\left(y z \right)}\right)$$

함수 $$$\sin{\left(y z \right)}$$$는 두 함수 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$와 $$$g{\left(z \right)} = y z$$$의 합성함수 $$$f{\left(g{\left(z \right)} \right)}$$$이다.

연쇄법칙 $$$\frac{d}{dz} \left(f{\left(g{\left(z \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dz} \left(g{\left(z \right)}\right)$$$을(를) 적용하십시오:

$${\color{red}\left(\frac{d}{dz} \left(\sin{\left(y z \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dz} \left(y z\right)\right)}$$

사인 함수의 도함수는 $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dz} \left(y z\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dz} \left(y z\right)$$

역치환:

$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dz} \left(y z\right) = \cos{\left({\color{red}\left(y z\right)} \right)} \frac{d}{dz} \left(y z\right)$$

상수배 법칙 $$$\frac{d}{dz} \left(c f{\left(z \right)}\right) = c \frac{d}{dz} \left(f{\left(z \right)}\right)$$$을 $$$c = y$$$와 $$$f{\left(z \right)} = z$$$에 적용합니다:

$$\cos{\left(y z \right)} {\color{red}\left(\frac{d}{dz} \left(y z\right)\right)} = \cos{\left(y z \right)} {\color{red}\left(y \frac{d}{dz} \left(z\right)\right)}$$

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

$$y \cos{\left(y z \right)} {\color{red}\left(\frac{d}{dz} \left(z\right)\right)} = y \cos{\left(y z \right)} {\color{red}\left(1\right)}$$

따라서, $$$\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right) = y \cos{\left(y z \right)}$$$.