Derivative of $$$\frac{1}{2 x}$$$
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Find $$$\frac{d}{dx} \left(\frac{1}{2 x}\right)$$$.
Solution
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 $$$c = \frac{1}{2}$$$ and $$$f{\left(x \right)} = \frac{1}{x}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\frac{1}{2 x}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dx} \left(\frac{1}{x}\right)}{2}\right)}$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = -1$$$:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(\frac{1}{x}\right)\right)}}{2} = \frac{{\color{red}\left(- \frac{1}{x^{2}}\right)}}{2}$$Thus, $$$\frac{d}{dx} \left(\frac{1}{2 x}\right) = - \frac{1}{2 x^{2}}$$$.
Answer
$$$\frac{d}{dx} \left(\frac{1}{2 x}\right) = - \frac{1}{2 x^{2}}$$$A