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