Question

How do you find the derivative of `7=xy-e^(xy)`?

Answer 1

`dy/dx = -y/x`

Explanation:

By implicit differentiation:

`d/dx(7) = d/dx(xy - e^(xy))`

`d/dx(7) = d/dx(xy) + d/dx(e^(xy))`

By using the identity `d/dx(e^(b(x))) = (b(x))' xx e^(b(x))`

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

`0 = y + x(dy/dx) + ye^(xy) + xe^(xy)(dy/dx)`

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

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

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

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

`dy/dx = -y/x`

Hopefully this helps!