1. The problem is to find the second derivative $f''(x)$ of the function $f(x) = 7x^3 - 6x^5$. 2. Recall the power rule for derivatives: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. 3. First, find the first derivative $f'(x)$: $$f'(x) = \frac{d}{dx}(7x^3) - \frac{d}{dx}(6x^5) = 7 \cdot 3x^{3-1} - 6 \cdot 5x^{5-1} = 21x^2 - 30x^4$$ 4. Next, find the second derivative $f''(x)$ by differentiating $f'(x)$: $$f''(x) = \frac{d}{dx}(21x^2) - \frac{d}{dx}(30x^4) = 21 \cdot 2x^{2-1} - 30 \cdot 4x^{4-1} = 42x - 120x^3$$ 5. Therefore, the second derivative is: $$f''(x) = 42x - 120x^3$$ This means the rate of change of the slope of the function $f(x)$ at any point $x$ is given by $42x - 120x^3$.