# What is the slope of the tangent line of  (xy-1/(yx))(xy+1/(xy))= C , where C is an arbitrary constant, at (1,5)?

Aug 8, 2016

$- 5$

#### Explanation:

$f \left(x , y\right) = \left(x y - \frac{1}{y x}\right) \left(x y + \frac{1}{x y}\right) - C = 0$ so

$\frac{\mathrm{dy}}{\mathrm{dx}} = - {f}_{x} / \left({f}_{y}\right) = \frac{\frac{2}{{x}^{3} {y}^{2}} + 2 x {y}^{2}}{\frac{2}{{x}^{2} {y}^{3}} + 2 {x}^{2} y} = - \frac{y}{x}$

In the point $\left\{x , y\right\} = \left\{1 , 5\right\}$ we have

$\frac{\mathrm{dy}}{\mathrm{dx}} = - 5$