1. Problem: Find the derivative of the function $f(x) = -5 \sqrt[3]{x^2}$. 2. Rewrite the function using exponents: $f(x) = -5 x^{\frac{2}{3}}$. 3. Use the power rule for derivatives: If $f(x) = ax^n$, then $f'(x) = a n x^{n-1}$. 4. Apply the power rule: $$f'(x) = -5 \times \frac{2}{3} x^{\frac{2}{3} - 1} = -\frac{10}{3} x^{-\frac{1}{3}}$$ 5. Simplify the expression: $$f'(x) = -\frac{10}{3} \frac{1}{x^{\frac{1}{3}}} = -\frac{10}{3} \frac{1}{\sqrt[3]{x}}$$ 6. Final answer: $$f'(x) = -\frac{10}{3 \sqrt[3]{x}}$$ This derivative tells us the rate of change of the function $f(x)$ at any point $x$, except where $x=0$ because of the negative exponent.