# What is the slope of the tangent line of e^(xy)-e^(x^2-x)/y = C , where C is an arbitrary constant, at (3,1)?

Apr 15, 2017

For` $f \left(x , y\right) = C$ we have the implicit function theorem:

$\frac{\mathrm{dy}}{\mathrm{dx}} = - {f}_{x} / {f}_{y}$

So:

${f}_{x} = y {e}^{x y} - \frac{1}{y} \left(2 x - 1\right) {e}^{{x}^{2} - x}$

And:

${f}_{y} = x {e}^{x y} + \frac{1}{y} ^ 2 {e}^{{x}^{2} - x}$

So:

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{y {e}^{x y} - \frac{1}{y} \left(2 x - 1\right) {e}^{{x}^{2} - x}}{x {e}^{x y} + \frac{1}{y} ^ 2 {e}^{{x}^{2} - x}}$

$= - \frac{{y}^{3} {e}^{x y} - y \left(2 x - 1\right) {e}^{{x}^{2} - x}}{x {y}^{2} {e}^{x y} + {e}^{{x}^{2} - x}}$

$\implies {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\left(3 , 1\right)} = - \frac{{e}^{3} - 5}{3 {e}^{3} + 1}$