1. The problem is to find the first and second derivatives of the function $f(x) = x^3 - 5x^2 + x - 1$. 2. Recall the power rule for derivatives: if $f(x) = x^n$, then $f'(x) = nx^{n-1}$. 3. Find the first derivative $f'(x)$ by differentiating each term: $$f'(x) = 3x^2 - 10x + 1$$ 4. Now find the second derivative $f''(x)$ by differentiating $f'(x)$: $$f''(x) = 6x - 10$$ 5. So, the first derivative is $f'(x) = 3x^2 - 10x + 1$ and the second derivative is $f''(x) = 6x - 10$. These derivatives tell us the slope of the function and the curvature respectively at any point $x$.