Derivative of $$$2 x^{2} - 3$$$
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Find $$$\frac{d}{dx} \left(2 x^{2} - 3\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dx} \left(2 x^{2} - 3\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(2 x^{2}\right) - \frac{d}{dx} \left(3\right)\right)}$$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 $$$c = 2$$$ and $$$f{\left(x \right)} = x^{2}$$$:
$${\color{red}\left(\frac{d}{dx} \left(2 x^{2}\right)\right)} - \frac{d}{dx} \left(3\right) = {\color{red}\left(2 \frac{d}{dx} \left(x^{2}\right)\right)} - \frac{d}{dx} \left(3\right)$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:
$$2 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} - \frac{d}{dx} \left(3\right) = 2 {\color{red}\left(2 x\right)} - \frac{d}{dx} \left(3\right)$$The derivative of a constant is $$$0$$$:
$$4 x - {\color{red}\left(\frac{d}{dx} \left(3\right)\right)} = 4 x - {\color{red}\left(0\right)}$$Thus, $$$\frac{d}{dx} \left(2 x^{2} - 3\right) = 4 x$$$.
Answer
$$$\frac{d}{dx} \left(2 x^{2} - 3\right) = 4 x$$$A