Derivative of $$$x + \sin{\left(x \right)}$$$
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Your Input
Find $$$\frac{d}{dx} \left(x + \sin{\left(x \right)}\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dx} \left(x + \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) + \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$The derivative of the sine is $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(x\right) = {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(x\right)$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = \cos{\left(x \right)} + {\color{red}\left(1\right)}$$Thus, $$$\frac{d}{dx} \left(x + \sin{\left(x \right)}\right) = \cos{\left(x \right)} + 1$$$.
Answer
$$$\frac{d}{dx} \left(x + \sin{\left(x \right)}\right) = \cos{\left(x \right)} + 1$$$A