Step 1: Write down the definition of the derivative from the first principle. The derivative of a function $f(x)$ is given by: $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} $$ Step 2: Identify $f(x)$ and $f(x+h)$. Given $y = f(x) = \frac{1}{\sqrt{x+1}}$. Then $f(x+h) = \frac{1}{\sqrt{(x+h)+1}} = \frac{1}{\sqrt{x+h+1}}$. Step 3: Substitute $f(x)$ and $f(x+h)$ into the limit definition. $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{\frac{1}{\sqrt{x+h+1}} - \frac{1}{\sqrt{x+1}}}{h} $$ Step 4: Simplify the numerator by finding a common denominator. $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{\frac{\sqrt{x+1} - \sqrt{x+h+1}}{\sqrt{x+h+1}\sqrt{x+1}}}{h} $$ $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{\sqrt{x+1} - \sqrt{x+h+1}}{h\sqrt{x+h+1}\sqrt{x+1}} $$ Step 5: Multiply the numerator and denominator by the conjugate of the numerator. The conjugate of $\sqrt{x+1} - \sqrt{x+h+1}$ is $\sqrt{x+1} + \sqrt{x+h+1}$. $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{\sqrt{x+1} - \sqrt{x+h+1}}{h\sqrt{x+h+1}\sqrt{x+1}} \times \frac{\sqrt{x+1} + \sqrt{x+h+1}}{\sqrt{x+1} + \sqrt{x+h+1}} $$ $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{(\sqrt{x+1})^2 - (\sqrt{x+h+1})^2}{h\sqrt{x+h+1}\sqrt{x+1}(\sqrt{x+1} + \sqrt{x+h+1})} $$ $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{(x+1) - (x+h+1)}{h\sqrt{x+h+1}\sqrt{x+1}(\sqrt{x+1} + \sqrt{x+h+1})} $$ $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{x+1-x-h-1}{h\sqrt{x+h+1}\sqrt{x+1}(\sqrt{x+1} + \sqrt{x+h+1})} $$ $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{-h}{h\sqrt{x+h+1}\sqrt{x+1}(\sqrt{x+1} + \sqrt{x+h+1})} $$ Step 6: Cancel out $h$ and evaluate the limit. $$ \frac{dy}{dx} = \lim_{h \to 0} \frac{-1}{\sqrt{x+h+1}\sqrt{x+1}(\sqrt{x+1} + \sqrt{x+h+1})} $$ Now, substitute $h=0$: $$ \frac{dy}{dx} = \frac{-1}{\sqrt{x+0+1}\sqrt{x+1}(\sqrt{x+1} + \sqrt{x+0+1})} $$ $$ \frac{dy}{dx} = \frac{-1}{\sqrt{x+1}\sqrt{x+1}(\sqrt{x+1} + \sqrt{x+1})} $$ $$ \frac{dy}{dx} = \frac{-1}{(x+1)(2\sqrt{x+1})} $$ $$ \frac{dy}{dx} = \frac{-1}{2(x+1)^{1}(x+1)^{1/2}} $$ $$ \frac{dy}{dx} = \frac{-1}{2(x+1)^{3/2}} $$ The derivative of $y = \frac{1}{\sqrt{x+1}}$ from the first principle is $\boxed{-\frac{1}{2(x+1)^{3/2}}}$.