Answer
$$\frac{{dy}}{{dx}} = - \frac{1}{{2{{\left( {x - 1} \right)}^{3/2}}}}$$
Work Step by Step
$$\eqalign{
& f\left( x \right) = \frac{1}{{\sqrt {x - 1} }} \cr
& {\text{Using the formula }}\frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{f\left( {x + \Delta x} \right) - f\left( x \right)}}{{\Delta x}} \cr
& \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\frac{1}{{\sqrt {x + \Delta x - 1} }} - \frac{1}{{\sqrt {x - 1} }}}}{{\Delta x}} \cr
& \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\frac{{\sqrt {x - 1} - \sqrt {x + \Delta x - 1} }}{{\sqrt {x - 1} \sqrt {x + \Delta x - 1} }}}}{{\Delta x}} \cr
& \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\sqrt {x - 1} - \sqrt {x + \Delta x - 1} }}{{\Delta x\sqrt {x - 1} \sqrt {x + \Delta x - 1} }} \times \frac{{\sqrt {x - 1} + \sqrt {x + \Delta x - 1} }}{{\sqrt {x - 1} + \sqrt {x + \Delta x - 1} }} \cr
& \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{{{\left( {\sqrt {x - 1} } \right)}^2} - {{\left( {\sqrt {x + \Delta x - 1} } \right)}^2}}}{{\Delta x\left( {\sqrt {x - 1} \sqrt {x + \Delta x - 1} } \right)\left( {\sqrt {x - 1} + \sqrt {x + \Delta x - 1} } \right)}} \cr
& \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{x - 1 - x - \Delta x + 1}}{{\Delta x\left( {\sqrt {x - 1} \sqrt {x + \Delta x - 1} } \right)\left( {\sqrt {x - 1} + \sqrt {x + \Delta x - 1} } \right)}} \cr
& \frac{{dy}}{{dx}} = \mathop {\lim }\limits_{\Delta x \to 0} \frac{{ - \Delta x}}{{\Delta x\left( {\sqrt {x - 1} \sqrt {x + \Delta x - 1} } \right)\left( {\sqrt {x - 1} + \sqrt {x + \Delta x - 1} } \right)}} \cr
& \frac{{dy}}{{dx}} = - \mathop {\lim }\limits_{\Delta x \to 0} \frac{1}{{\left( {\sqrt {x - 1} \sqrt {x + \Delta x - 1} } \right)\left( {\sqrt {x - 1} + \sqrt {x + \Delta x - 1} } \right)}} \cr
& \cr
& \Delta x \to 0 \cr
& \frac{{dy}}{{dx}} = - \frac{1}{{\left( {\sqrt {x - 1} \sqrt {x + 0 - 1} } \right)\left( {\sqrt {x - 1} + \sqrt {x + 0 - 1} } \right)}} \cr
& \frac{{dy}}{{dx}} = - \frac{1}{{\left( {x - 1} \right)\left( {2\sqrt {x - 1} } \right)}} \cr
& \frac{{dy}}{{dx}} = - \frac{1}{{2{{\left( {x - 1} \right)}^{3/2}}}} \cr} $$