Single Variable Calculus, Chapter 3, 3.3, Section 3.3, Problem 58

Find the 1st and 2nd derivatives of $G(r) = \sqrt{r} + \sqrt[3]{r}$


$\begin{equation} \begin{aligned} G'(r) =& \frac{d}{dr} (r^{\frac{1}{2}}) + \frac{d}{dr} (r^{\frac{1}{3}}) && \text{Derive each term} \\ \\ G'(r) =& \frac{(r)^{\frac{-1}{2}}}{2} + \frac{(r)^{\frac{-2}{3}}}{3} && \text{Simplify the equation} \\ \\ G'(r) =& \frac{1}{2 \sqrt{r}} + \frac{1}{3 \sqrt[3]{r^2}} && \text{1st derivative of $G(r)$} \\ \\ \\ \\ G''(r) =& \frac{(2)(r)^{\frac{1}{2}} \displaystyle \frac{d}{dr} (1) - \left[ (1)(2) \frac{d}{dr} (r)^{\frac{1}{2}} \right] }{(2 \sqrt{r})^2} + \frac{\displaystyle 3 (r^{\frac{2}{3}}) \frac{d}{dr} (1) - \left[ (1)(3) \frac{d}{dr} (r^{\frac{2}{3}}) \right]}{(3 \sqrt[3]{r^2})^2} && \text{Using Quotient Rule} \\ \\ G''(r) =& \frac{\displaystyle (2)(r^{\frac{1}{2}}) (0) - (1)(2)\left( \frac{1}{2} r^{\frac{-1}{2}} \right) }{4r} + \frac{\displaystyle (3)(r^{\frac{2}{3}}) (0) - (1)(3) \left( \frac{2}{3} r^{\frac{-1}{3}} \right)}{9 \sqrt[3]{r^4}} && \text{Simplify the equation} \\ \\ G''(r) =& \frac{r^{\frac{-1}{2}}}{4r} - \frac{2r^{\frac{-1}{3}}}{9 \sqrt[3]{r^4}} && \text{Simplify the equation} \\ \\ G''(r) =& \frac{1}{4r \sqrt{r}} - \frac{2}{9 r \sqrt[3]{r^2}} && \text{2nd derivative of $G(r)$} \\ \\ \end{aligned} \end{equation}$