Derivative of $$$1 - \cos{\left(x \right)}$$$
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Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dx} \left(1 - \cos{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(1\right) - \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$The derivative of a constant is $$$0$$$:
$${\color{red}\left(\frac{d}{dx} \left(1\right)\right)} - \frac{d}{dx} \left(\cos{\left(x \right)}\right) = {\color{red}\left(0\right)} - \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$The derivative of the cosine is $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = - {\color{red}\left(- \sin{\left(x \right)}\right)}$$Thus, $$$\frac{d}{dx} \left(1 - \cos{\left(x \right)}\right) = \sin{\left(x \right)}$$$.
Answer
$$$\frac{d}{dx} \left(1 - \cos{\left(x \right)}\right) = \sin{\left(x \right)}$$$A