Question

$$ \frac { d r } { d s } = - \frac { 2 } { 3 s ^ { 3 } } + \frac { 24 } { 5 s ^ { 5 } } $$ $$ \frac { d ^ { 2 } r } { d s ^ { 2 } } = square $$

Explanation:

Step1: Rewrite the function

Rewrite \(\frac{dr}{ds}=-\frac{2}{3s^{3}}+\frac{24}{5s^{5}}\) as \(\frac{dr}{ds}=-\frac{2}{3}s^{- 3}+\frac{24}{5}s^{-5}\).

Step2: Differentiate using the power rule

The power rule is \(\frac{d}{dx}(x^{n})=nx^{n - 1}\).
Differentiate \(-\frac{2}{3}s^{-3}\): \(\frac{d}{ds}(-\frac{2}{3}s^{-3})=(-\frac{2}{3})\times(-3)s^{-3 - 1}=2s^{-4}\).
Differentiate \(\frac{24}{5}s^{-5}\): \(\frac{d}{ds}(\frac{24}{5}s^{-5})=\frac{24}{5}\times(-5)s^{-5 - 1}=-24s^{-6}\).

Answer:

\(\frac{d^{2}r}{ds^{2}}=\frac{2}{s^{4}}-\frac{24}{s^{6}}\)