# Second derivative of $x^{3}$

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

Related calculators: Derivative Calculator, Logarithmic Differentiation Calculator

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

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

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