Question

Find the first and second derivatives of the function.

50. \(G\left( r \right) = \sqrt r + \sqrt(3){r}\)

Short answer

The first derivative is \(G'\left( r \right) = \frac{1}{2}{r^{ - 1/2}} + \frac{1}{3}{r^{ - 2/3}}\)and the second derivative is \(G''\left( r \right) = - \frac{1}{4}{r^{ - 3/2}} - \frac{2}{9}{r^{ - 5/3}}\).

Step-by-step solution

Find the first derivative

The given function is \(G\left( r \right) = \sqrt r + \sqrt(3){r}\).

Use sum rule of differentiation which implies that \(\frac{d}{{dx}}\left( {f\left( x \right) + g\left( x \right)} \right) = \frac{d}{{dx}}f\left( x \right) + \frac{d}{{dx}}g\left( x \right)\) and power rule of differentiation which implies that \(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\), where \(n\) is any real number.

On differentiating this function with respect to \(r\) by using sum and power rule, the following expression is obtained:

\(\begin{aligned}G'\left( r \right) &= \frac{d}{{dr}}\left( {\sqrt r + \sqrt(3){r}} \right)\\ &= \frac{d}{{dr}}\left( {\sqrt r } \right) + \frac{d}{{dr}}\left( {\sqrt(3){r}} \right)\\ &= \frac{1}{2}{r^{ - 1/2}} + \frac{1}{3}{r^{ - 2/3}}\end{aligned}\)

Thus, the first derivative of the function is \(G'\left( r \right) = \frac{1}{2}{r^{ - 1/2}} + \frac{1}{3}{r^{ - 2/3}}\).

Find the second derivative

The first derivative of the function is \(G'\left( r \right) = \frac{1}{2}{r^{ - 1/2}} + \frac{1}{3}{r^{ - 2/3}}\). On differentiating it again, with respect to \(r\) axis by using sum and power rule, we obtain the second derivative, as follows:

\(\begin{aligned}G''\left( r \right) &= \frac{d}{{dr}}\left( {\frac{1}{2}{r^{ - 1/2}} + \frac{1}{3}{r^{ - 2/3}}} \right)\\ &= \frac{d}{{dr}}\left( {\frac{1}{2}{r^{ - 1/2}}} \right) + \frac{d}{{dr}}\left( {\frac{1}{3}{r^{ - 2/3}}} \right)\\ &= \frac{1}{2}\left( { - \frac{1}{2}} \right){r^{ - 3/2}} + \frac{1}{3}\left( { - \frac{2}{3}} \right){r^{ - 5/3}}\\ &= - \frac{1}{4}{r^{ - 3/2}} - \frac{2}{9}{r^{ - 5/3}}\end{aligned}\)

Thus, the second derivative is \(G''\left( r \right) = - \frac{1}{4}{r^{ - 3/2}} - \frac{2}{9}{r^{ - 5/3}}\).