Question

Differentiate the function.

\(y = \frac{{s - \sqrt s }}{{{s^2}}}\)

Short answer

Derivative of the function is \(\frac{{3 - 2\sqrt s }}{{2{s^{5/2}}}}\).

Step-by-step solution

Precise Definition of differentiation

A derivative is used in calculus mathematics. It is the rate of change of a function with respect to a variable. A derivative is a function of a single variable at a selected input value.

Find the derivative of y using the Quotient rule

The equation for the Quotient rule is \({\left( {\frac{f}{g}} \right)^\prime } = \frac{{gf' - fg'}}{{{{\left( g \right)}^2}}}\).

Apply Quotient rule for the function.

\(\begin{aligned}\frac{{dy}}{{ds}} &= \frac{d}{{ds}}\left( {\frac{{s - \sqrt s }}{{{s^2}}}} \right)\\ &= \frac{{{s^2}\frac{d}{{ds}}\left( {s - \sqrt s } \right) - \left( {s - \sqrt s } \right)\frac{d}{{ds}}\left( {{s^2}} \right)}}{{{{\left( {{s^2}} \right)}^2}}}\end{aligned}\)

Simplify the obtained condition

Simplify the obtained derivative.

\(\begin{aligned}{y}'& =\frac{{{s}^{2}}\frac{d}{ds}\left( s-\sqrt{s} \right)-\left( s-\sqrt{s} \right)\frac{d}{ds}\left( {{s}^{2}} \right)}{{{s}^{4}}} \\ & =\frac{{{s}^{2}}\left( 1-\frac{1}{2\sqrt{s}} \right)-\left( s-\sqrt{s} \right)\left( 2s \right)}{{{s}^{4}}} \\ & =\frac{\left( {{s}^{2}}-\frac{{{s}^{3/2}}}{2} \right)-\left( 2{{s}^{2}}-2{{s}^{{}^{3}/{}_{2}}} \right)}{{{s}^{4}}} \\ & =\frac{-{{s}^{2}}+\frac{3}{2}{{s}^{3/2}}}{{{s}^{4}}} \\ & =-\frac{1}{{{s}^{2}}}+\frac{3}{2{{s}^{5/2}}} \\ & =\frac{3-2\sqrt{s}}{2{{s}^{5/2}}} \end{aligned}\)

So, \(y' = \frac{{3 - 2\sqrt s }}{{2{s^{5/2}}}}\) is the final answer of this derivative.