1. **State the problem:** We are given the function $f(x) = \frac{4}{5x^5}$ and need to find its derivative $f'(x)$. 2. **Rewrite the function:** Express $f(x)$ with negative exponents for easier differentiation: $$f(x) = \frac{4}{5} x^{-5}$$ 3. **Apply the power rule:** The derivative of $x^n$ is $nx^{n-1}$. So, $$f'(x) = \frac{4}{5} \cdot (-5) x^{-5-1} = \frac{4}{5} \cdot (-5) x^{-6}$$ 4. **Simplify the constants:** $$\frac{4}{5} \cdot (-5) = -4$$ 5. **Write the final derivative:** $$f'(x) = -4 x^{-6} = -\frac{4}{x^6}$$ **Answer:** $$f'(x) = -\frac{4}{x^6}$$