If x^2 + 4x + 3xy + y^3 = 6, find the first derivative.

This is an example of implicit differentiation. We differentiate with respect to x, remembering that y is a function of x. The terms without a 'y' differentiate as normal by the power rule nx^n-1. This is the terms x², 4x and the 6 on the other side of the = sign which goes to zero as it is a constant. y differentiates to dy/dx so the product rule is used on the term 3xy and the chain rule is used on the term y³. 2x + 4 + 3x*dy/dx + 3y + 3y²dy/dx = 0 As dy/dx is a product in two of the terms, we can factorise this out. dy/dx(3x + 3y²) = -(2x +3y + 4)dy/dx = -(2x + 3y + 4)/(3x + 3y²)