Question

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)`?

Answer 1

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

`dy/dx = - f_x/f_y`

So:

`f_x = y e^(xy) - 1/ y(2x - 1) e^(x^2 - x)`

And:

`f_y = x e^(xy) + 1/y^2 e^(x^2 - x)`

So:

`dy/dx = - (y e^(xy) - 1/ y(2x - 1) e^(x^2 - x))/(x e^(xy) + 1/y^2 e^(x^2 - x))`

`= - (y^3 e^(xy) - y(2x - 1) e^(x^2 - x))/(xy^2 e^(xy) + e^(x^2 - x))`

`implies (dy/dx)_{ (3,1)} = - (e^(3) - 5 )/(3 e^(3) + 1)`