1. Problem: Differentiate the function $f(x) = 2x^3 + 6x$. 2. Formula: Use the power rule for differentiation, which states $\frac{d}{dx} x^n = nx^{n-1}$. 3. Apply the power rule to each term: $$\frac{d}{dx} (2x^3) = 2 \cdot 3x^{3-1} = 6x^2$$ $$\frac{d}{dx} (6x) = 6 \cdot 1x^{1-1} = 6$$ 4. Combine the derivatives: $$f'(x) = 6x^2 + 6$$ 5. Explanation: We differentiated each term separately using the power rule and then added the results to get the derivative of the entire function. Final answer: $f'(x) = 6x^2 + 6$.