There isn’t much to do here other than take the derivative using the rules we discussed in this section.

Remember that you’ll need to rewrite the terms so that each of the \(z\)’s are in the numerator with negative exponents and rewrite the root as a fractional exponent before taking the derivative. Here is the rewritten function.

\[R\left( z \right) = 6{z^{ - \,\,\frac{3}{2}}} + \frac{1}{8}{z^{ - 4}} - \frac{1}{3}{z^{ - 10}}\]

The derivative is,

\[R'\left( z \right) = 6\left( { - \frac{3}{2}} \right){z^{ - \,\,\frac{5}{2}}} + \frac{1}{8}\left( { - 4} \right){z^{ - 5}} - \frac{1}{3}\left( { - 10} \right){z^{ - 11}} = \require{bbox} \bbox[2pt,border:1px solid black]{{ - 9{z^{ - \,\,\frac{5}{2}}} - \frac{1}{2}{z^{ - 5}} + \frac{{10}}{3}{z^{ - 11}}}}\]