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Second derivative of $$$x^{3}$$$

The calculator will find the second derivative of $$$x^{3}$$$, with steps shown.

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Your Input

Find $$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right)$$$.

Solution

Find the first derivative $$$\frac{d}{dx} \left(x^{3}\right)$$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 3$$$:

$${\color{red}\left(\frac{d}{dx} \left(x^{3}\right)\right)} = {\color{red}\left(3 x^{2}\right)}$$

Thus, $$$\frac{d}{dx} \left(x^{3}\right) = 3 x^{2}$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right) = \frac{d}{dx} \left(3 x^{2}\right)$$$

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 $$$f{\left(x \right)} = x^{2}$$$:

$${\color{red}\left(\frac{d}{dx} \left(3 x^{2}\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(x^{2}\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$$3 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = 3 {\color{red}\left(2 x\right)}$$

Thus, $$$\frac{d}{dx} \left(3 x^{2}\right) = 6 x$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right) = 6 x$$$.

Answer

$$$\frac{d^{2}}{dx^{2}} \left(x^{3}\right) = 6 x$$$A