Derivative of $$$x e^{8} - 9$$$
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Find $$$\frac{d}{dx} \left(x e^{8} - 9\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dx} \left(x e^{8} - 9\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x e^{8}\right) - \frac{d}{dx} \left(9\right)\right)}$$The derivative of a constant is $$$0$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(9\right)\right)} + \frac{d}{dx} \left(x e^{8}\right) = - {\color{red}\left(0\right)} + \frac{d}{dx} \left(x e^{8}\right)$$Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = e^{8}$$$ and $$$f{\left(x \right)} = x$$$:
$${\color{red}\left(\frac{d}{dx} \left(x e^{8}\right)\right)} = {\color{red}\left(e^{8} \frac{d}{dx} \left(x\right)\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$e^{8} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = e^{8} {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{dx} \left(x e^{8} - 9\right) = e^{8}$$$.
Answer
$$$\frac{d}{dx} \left(x e^{8} - 9\right) = e^{8}$$$A