The curve C has equation 4x^2 – y^3 – 4xy + 2^y = 0 The point P with coordinates (–2, 4) lies on C . Find the exact value of dy/dx at the point P .
Since we need to find dy/dx, we must first differentiate the equation implicitly which gives us: 8x - 3y2dy/dx - 4y - 4xdy/dx + 2yln(2)dy/dx = 0. Because we are given a point, we can substitute in the x and y values of that point which results in: -16 - 48dy/dx - 16 + 8dy/dx + 16ln(2)dy/dx = 0.We now have an equation which is easily solved by rearrangement. First we bring all dy/dx's to one side: 16ln(2)dy/dx - 40dy/dx = 32. And then we isolate dy/dx: dy/dx(16ln(2) - 40) = 32 => dy/dx = 32/(16ln(2) - 40).
