QUESTION 3 (16 manks) Find the derivative for the following functions, giving your answers in their sinplast form. a. \( y=4 x^{3}-2 \sqrt{x}-\frac{1}{x^{3}} \)

To find the derivative of \( y=4 x^{3}-2 \sqrt{x}-\frac{1}{x^{3}} \), we apply the power rule, the derivative of a square root, and the derivative of \( x^{-n} \). 1. The derivative of \( 4 x^{3} \) is \( 12 x^{2} \). 2. The derivative of \( -2 \sqrt{x} \) is \( -\frac{2}{2\sqrt{x}} = -\frac{1}{\sqrt{x}} \) or simplified, \( -x^{-\frac{1}{2}} \). 3. The derivative of \( -\frac{1}{x^{3}} \) is \( 3x^{-4} \). Now put it all together: \[ \frac{dy}{dx} = 12 x^{2} - \frac{1}{\sqrt{x}} + 3x^{-4} \] Or in simplest form: \[ \frac{dy}{dx} = 12 x^{2} - x^{-\frac{1}{2}} + \frac{3}{x^{4}} \]