The curve C has equation (4x^2-y^3+3^2x)=0. The point P (0,1) lies on C: what is the value of dy/dx at P?

Use the chain rule to differentiate the original equation: this results in 8x-3y^2*(dy/dx) + 2ln(3)3^2x=0. This can be rearranged to find dy/dx as a function of y and x: 3y^2(dy/dx)=8x+2ln(3)*3^2x -> dy/dx=(8x+2ln(3)3^2x)/3y^2. At this point, dy/dx at point P can be computed: dy/dx=(80+2ln(3)3^0)/31^2=2ln(3)/3