1. The problem is to find the second derivative of the function $f(x) = x^3$. 2. Recall that the first derivative of a function $f(x)$, denoted $f'(x)$ or $\frac{d}{dx}f(x)$, gives the rate of change of the function. 3. The second derivative, denoted $f''(x)$ or $\frac{d^2}{dx^2}f(x)$, is the derivative of the first derivative and gives the rate of change of the rate of change. 4. Start by finding the first derivative of $f(x) = x^3$ using the power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ 5. Applying the power rule: $$f'(x) = 3x^{3-1} = 3x^2$$ 6. Now find the second derivative by differentiating $f'(x) = 3x^2$ again: $$f''(x) = \frac{d}{dx} 3x^2 = 3 \cdot 2 x^{2-1} = 6x$$ 7. Therefore, the second derivative of $x^3$ is $6x$. This means the curvature or concavity of the function $x^3$ at any point $x$ is given by $6x$.