1. Problem: Calculate the derivative of the function \(f(x) = -5x^6 + 4x^3 - \frac{1}{2}x + 4\). 2. Formula: The derivative of a power function \(x^n\) is given by \(\frac{d}{dx}x^n = nx^{n-1}\). 3. Apply the derivative term-by-term: - Derivative of \(-5x^6\) is \(-5 \times 6x^{6-1} = -30x^5\). - Derivative of \(4x^3\) is \(4 \times 3x^{3-1} = 12x^2\). - Derivative of \(-\frac{1}{2}x\) is \(-\frac{1}{2} \times 1x^{1-1} = -\frac{1}{2}\). - Derivative of constant \(4\) is \(0\). 4. Combine all derivatives: $$f'(x) = -30x^5 + 12x^2 - \frac{1}{2}$$ 5. Explanation: We used the power rule for derivatives, which states that to differentiate \(x^n\), multiply by the exponent and reduce the exponent by one. Constants multiply as usual, and the derivative of a constant is zero. Final answer: $$f'(x) = -30x^5 + 12x^2 - \frac{1}{2}$$