Question

How do you find the derivative of `5=3e^(xy)+x^2y+xy^2`?

Answer 1

`dy/dx = -(3ye^x+2xy+y^2)/(3e^x+ x^2+ 2xy)`

Explanation:

We need to use implicit differentiation and the product rule;
`d/dx(uv)=u(dv)/dx+v(du)/dx`

`5=3e^xy+x^2y+xy^2`

`:. 0=3{e^xd/dxy+yd/dxe^x} + {x^2d/dxy+yd/dxx^2} + {xd/dyy^2+y^2d/dxx}`

`:. 0=3{e^xdy/dx+ye^x} + {x^2dy/dx+y(2x)} + {x(2ydy/dx)+y^2}`

`:. 0=3e^xdy/dx+3ye^x + x^2dy/dx+2xy + 2xydy/dx+y^2`
`:. {3e^x+ x^2+ 2xy}dy/dx = -(3ye^x+2xy+y^2)`
`:. dy/dx = -(3ye^x+2xy+y^2)/(3e^x+ x^2+ 2xy)`