Given that y = 4x^{3} - 1 + 2x^{rac{1}{2}}, x > 0, find \frac{dy}{dx}. - Edexcel - A-Level Maths Pure - Question 3 - 2007 - Paper 2

To find \frac{dy}{dx}, we differentiate each term in the equation:\n\n1. For the term 4x^{3}, the derivative is: \n [ 4 \cdot 3x^{2} = 12x^{2} ] \n\n2. For the term -1, the derivative is 0 since it's a constant. \n\n3. For the term 2x^{\frac{1}{2}}, we apply the power rule: \n [ 2 \cdot \frac{1}{2}x^{-\frac{1}{2}} = x^{-\frac{1}{2}} ] \n\nCombining these results, we get: \n[ \frac{dy}{dx} = 12x^{2} + x^{-\frac{1}{2}} ]