1. **State the problem:** We need to find the derivative of the function $$y = 5x^6 - \sec^{-1}(x)$$. 2. **Recall the derivative rules:** - The derivative of $$x^n$$ is $$nx^{n-1}$$. - The derivative of $$\sec^{-1}(x)$$ is $$\frac{1}{|x|\sqrt{x^2 - 1}}$$ for $$|x| > 1$$. 3. **Differentiate each term:** - For $$5x^6$$, the derivative is $$5 \times 6x^{5} = 30x^{5}$$. - For $$-\sec^{-1}(x)$$, the derivative is $$- \frac{1}{|x|\sqrt{x^2 - 1}}$$. 4. **Combine the results:** $$\frac{dy}{dx} = 30x^{5} - \frac{1}{|x|\sqrt{x^2 - 1}}$$. 5. **Interpretation:** The derivative tells us the rate of change of the function at any point $$x$$ where $$|x| > 1$$ (domain restriction for $$\sec^{-1}(x)$$).