Question

How do you implicitly differentiate `y= y^2 + e^(x y)`?

Answer 1

We use the power rule and the rule if `y= e^(f(x)`, then its derivative `y' = f'(x)e^(f(x))`.

Hence:

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

Solving for `dy/dx`:

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

`dy/dx - 2y(dy/dx) - xe^(xy)(dy/dx) = ye^(xy)`

`dy/dx(1 - 2y - xe^(xy)) = ye^(xy)`

`dy/dx = (ye^(xy))/(1 - 2y - xe^(xy))`

Hopefully this helps!