Question

find the first and second derivatives.

$r = \frac{1}{3s^{2}} - \frac{6}{5s^{4}}$

$\frac{dr}{ds} = \square$

Explanation:

Step1: Rewrite the function

Rewrite \(r=\frac{1}{3s^{2}}-\frac{6}{5s^{4}}\) as \(r = \frac{1}{3}s^{- 2}-\frac{6}{5}s^{-4}\).

Step2: Find the first - derivative using the power rule

The power rule is \(\frac{d}{ds}(s^{n})=ns^{n - 1}\).
For the first term: \(\frac{d}{ds}(\frac{1}{3}s^{-2})=\frac{1}{3}\times(-2)s^{-2 - 1}=-\frac{2}{3}s^{-3}\).
For the second term: \(\frac{d}{ds}(-\frac{6}{5}s^{-4})=-\frac{6}{5}\times(-4)s^{-4 - 1}=\frac{24}{5}s^{-5}\).
So, \(\frac{dr}{ds}=-\frac{2}{3s^{3}}+\frac{24}{5s^{5}}\).

Answer:

\(-\frac{2}{3s^{3}}+\frac{24}{5s^{5}}\)