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