10. Demonstrate and explain how to find the derivative of the following functions. Be sure to explicitly denote which derivative rules (scalar multiple, sum/difference, quotient, etc.) you are using in your work. \( \begin{array}{l}\text { (g) } \\ f(w)=\frac{6 w^{2}-w-3}{w^{3}}\end{array} \)

To find the derivative of the function \( f(w)=\frac{6 w^{2}-w-3}{w^{3}} \), we can apply the quotient rule. The quotient rule states that if you have a function expressed as \( \frac{u(w)}{v(w)} \), its derivative is given by \( f'(w) = \frac{u'(w)v(w) - u(w)v'(w)}{(v(w))^2} \). Here, let \( u(w) = 6w^2 - w - 3 \) and \( v(w) = w^3 \). First, we find the derivatives: \( u'(w) = 12w - 1 \) and \( v'(w) = 3w^2 \). Applying the quotient rule, we have: \[ f'(w) = \frac{(12w - 1)(w^3) - (6w^2 - w - 3)(3w^2)}{(w^3)^2} \] This simplifies down to \( f'(w) = \frac{(12w - 1)w^3 - (18w^2 - 3w - 9)w^2}{w^6} \). Now you can further simplify the numerator for the final answer. By carefully breaking the pieces down using the quotient rule, you've got yourself a solid foundation for dealing with derivatives of rational functions! Happy differentiating!