Find $$$\frac{d^{2}}{dx^{2}} \left(\sqrt{x} + \frac{5}{\sqrt{x}}\right)$$$

Find the first derivative $$$\frac{d}{dx} \left(\sqrt{x} + \frac{5}{\sqrt{x}}\right)$$$

The derivative of a sum/difference is the sum/difference of derivatives:

$${\color{red}\left(\frac{d}{dx} \left(\sqrt{x} + \frac{5}{\sqrt{x}}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\sqrt{x}\right) + \frac{d}{dx} \left(\frac{5}{\sqrt{x}}\right)\right)}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = \frac{1}{2}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sqrt{x}\right)\right)} + \frac{d}{dx} \left(\frac{5}{\sqrt{x}}\right) = {\color{red}\left(\frac{1}{2 \sqrt{x}}\right)} + \frac{d}{dx} \left(\frac{5}{\sqrt{x}}\right)$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = 5$$$ and $$$f{\left(x \right)} = \frac{1}{\sqrt{x}}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\frac{5}{\sqrt{x}}\right)\right)} + \frac{1}{2 \sqrt{x}} = {\color{red}\left(5 \frac{d}{dx} \left(\frac{1}{\sqrt{x}}\right)\right)} + \frac{1}{2 \sqrt{x}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = - \frac{1}{2}$$$:

$$5 {\color{red}\left(\frac{d}{dx} \left(\frac{1}{\sqrt{x}}\right)\right)} + \frac{1}{2 \sqrt{x}} = 5 {\color{red}\left(- \frac{1}{2 x^{\frac{3}{2}}}\right)} + \frac{1}{2 \sqrt{x}}$$

Simplify:

$$\frac{1}{2 \sqrt{x}} - \frac{5}{2 x^{\frac{3}{2}}} = \frac{x - 5}{2 x^{\frac{3}{2}}}$$

Thus, $$$\frac{d}{dx} \left(\sqrt{x} + \frac{5}{\sqrt{x}}\right) = \frac{x - 5}{2 x^{\frac{3}{2}}}$$$.

Next, $$$\frac{d^{2}}{dx^{2}} \left(\sqrt{x} + \frac{5}{\sqrt{x}}\right) = \frac{d}{dx} \left(\frac{x - 5}{2 x^{\frac{3}{2}}}\right)$$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = \frac{1}{2}$$$ and $$$f{\left(x \right)} = \frac{x - 5}{x^{\frac{3}{2}}}$$$:

$${\color{red}\left(\frac{d}{dx} \left(\frac{x - 5}{2 x^{\frac{3}{2}}}\right)\right)} = {\color{red}\left(\frac{\frac{d}{dx} \left(\frac{x - 5}{x^{\frac{3}{2}}}\right)}{2}\right)}$$

Apply the quotient rule $$$\frac{d}{dx} \left(\frac{f{\left(x \right)}}{g{\left(x \right)}}\right) = \frac{\frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} - f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)}{g^{2}{\left(x \right)}}$$$ with $$$f{\left(x \right)} = x - 5$$$ and $$$g{\left(x \right)} = x^{\frac{3}{2}}$$$:

$$\frac{{\color{red}\left(\frac{d}{dx} \left(\frac{x - 5}{x^{\frac{3}{2}}}\right)\right)}}{2} = \frac{{\color{red}\left(\frac{\frac{d}{dx} \left(x - 5\right) x^{\frac{3}{2}} - \left(x - 5\right) \frac{d}{dx} \left(x^{\frac{3}{2}}\right)}{\left(x^{\frac{3}{2}}\right)^{2}}\right)}}{2}$$

The derivative of a sum/difference is the sum/difference of derivatives:

$$\frac{x^{\frac{3}{2}} {\color{red}\left(\frac{d}{dx} \left(x - 5\right)\right)} - \left(x - 5\right) \frac{d}{dx} \left(x^{\frac{3}{2}}\right)}{2 x^{3}} = \frac{x^{\frac{3}{2}} {\color{red}\left(\frac{d}{dx} \left(x\right) - \frac{d}{dx} \left(5\right)\right)} - \left(x - 5\right) \frac{d}{dx} \left(x^{\frac{3}{2}}\right)}{2 x^{3}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$\frac{x^{\frac{3}{2}} \left({\color{red}\left(\frac{d}{dx} \left(x\right)\right)} - \frac{d}{dx} \left(5\right)\right) - \left(x - 5\right) \frac{d}{dx} \left(x^{\frac{3}{2}}\right)}{2 x^{3}} = \frac{x^{\frac{3}{2}} \left({\color{red}\left(1\right)} - \frac{d}{dx} \left(5\right)\right) - \left(x - 5\right) \frac{d}{dx} \left(x^{\frac{3}{2}}\right)}{2 x^{3}}$$

The derivative of a constant is $$$0$$$:

$$\frac{x^{\frac{3}{2}} \left(1 - {\color{red}\left(\frac{d}{dx} \left(5\right)\right)}\right) - \left(x - 5\right) \frac{d}{dx} \left(x^{\frac{3}{2}}\right)}{2 x^{3}} = \frac{x^{\frac{3}{2}} \left(1 - {\color{red}\left(0\right)}\right) - \left(x - 5\right) \frac{d}{dx} \left(x^{\frac{3}{2}}\right)}{2 x^{3}}$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = \frac{3}{2}$$$:

$$\frac{x^{\frac{3}{2}} - \left(x - 5\right) {\color{red}\left(\frac{d}{dx} \left(x^{\frac{3}{2}}\right)\right)}}{2 x^{3}} = \frac{x^{\frac{3}{2}} - \left(x - 5\right) {\color{red}\left(\frac{3 \sqrt{x}}{2}\right)}}{2 x^{3}}$$

Simplify:

$$\frac{x^{\frac{3}{2}} - \frac{3 \sqrt{x} \left(x - 5\right)}{2}}{2 x^{3}} = \frac{15 - x}{4 x^{\frac{5}{2}}}$$

Thus, $$$\frac{d}{dx} \left(\frac{x - 5}{2 x^{\frac{3}{2}}}\right) = \frac{15 - x}{4 x^{\frac{5}{2}}}$$$.

Therefore, $$$\frac{d^{2}}{dx^{2}} \left(\sqrt{x} + \frac{5}{\sqrt{x}}\right) = \frac{15 - x}{4 x^{\frac{5}{2}}}$$$.