Derivative of $$$4 - 3 x^{2}$$$
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Find $$$\frac{d}{dx} \left(4 - 3 x^{2}\right)$$$.
Solution
The derivative of a sum/difference is the sum/difference of derivatives:
$${\color{red}\left(\frac{d}{dx} \left(4 - 3 x^{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(4\right) - \frac{d}{dx} \left(3 x^{2}\right)\right)}$$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 $$$c = 3$$$ and $$$f{\left(x \right)} = x^{2}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) = - {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(4\right)$$Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:
$$- 3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(4\right) = - 3 {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(4\right)$$The derivative of a constant is $$$0$$$:
$$- 6 x + {\color{red}\left(\frac{d}{dx} \left(4\right)\right)} = - 6 x + {\color{red}\left(0\right)}$$Thus, $$$\frac{d}{dx} \left(4 - 3 x^{2}\right) = - 6 x$$$.
Answer
$$$\frac{d}{dx} \left(4 - 3 x^{2}\right) = - 6 x$$$A