$$$x^{2} + 6 x + 25$$$의 도함수

합/차의 도함수는 도함수들의 합/차이다:

$${\color{red}\left(\frac{d}{dx} \left(x^{2} + 6 x + 25\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{2}\right) + \frac{d}{dx} \left(6 x\right) + \frac{d}{dx} \left(25\right)\right)}$$

상수의 도함수는 $$$0$$$입니다:

$${\color{red}\left(\frac{d}{dx} \left(25\right)\right)} + \frac{d}{dx} \left(6 x\right) + \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(0\right)} + \frac{d}{dx} \left(6 x\right) + \frac{d}{dx} \left(x^{2}\right)$$

상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = 6$$$와 $$$f{\left(x \right)} = x$$$에 적용합니다:

$${\color{red}\left(\frac{d}{dx} \left(6 x\right)\right)} + \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(6 \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(x^{2}\right)$$

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

$$6 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(x^{2}\right) = 6 {\color{red}\left(1\right)} + \frac{d}{dx} \left(x^{2}\right)$$

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

$${\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + 6 = {\color{red}\left(2 x\right)} + 6$$

따라서, $$$\frac{d}{dx} \left(x^{2} + 6 x + 25\right) = 2 x + 6$$$.