# How do you differentiate y= sqrt(x/(x-9))?

May 2, 2017

(dy)/(dx)=-9/(2(x-9)sqrt(x(x-9))

#### Explanation:

Quotient rule states if $y \left(x\right) = \frac{g \left(x\right)}{h \left(x\right)}$

then $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{\mathrm{dg}}{\mathrm{dx}} \times h \left(x\right) - \frac{\mathrm{dh}}{\mathrm{dx}} \times g \left(x\right)}{h \left(x\right)} ^ 2$

Hence as $y + \sqrt{\frac{x}{x - 9}} = \frac{\sqrt{x}}{\sqrt{x - 9}}$

So here $g \left(x\right) = \sqrt{x}$ and $\frac{\mathrm{dg}}{\mathrm{dx}} = \frac{1}{2 \sqrt{x}}$

and $h \left(x\right) = \sqrt{x - 9}$ and $\frac{\mathrm{dh}}{\mathrm{dx}} = \frac{1}{2 \sqrt{x - 9}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\frac{1}{2 \sqrt{x}} \times \sqrt{x - 9} - \frac{1}{2 \sqrt{x - 9}} \times \sqrt{x}}{x - 9}$

=((x-9)/2-x/2)/((x-9)sqrt(x(x-9))

=-9/(2(x-9)sqrt(x(x-9))