Use Implicit Differentiation to find dy/dx of the following equation: 3(x)^2 + 8xy + 5(y)^2 = 4

(This is a common question in C4, the "hardest" module of the old A level)When differentiating we consider each term one at a time:Differentiating 3x² simply gives us 6x Differentiating 8xy is harder, we have to use the product rule: that is for any functions f and g: d(fg)/dx = f dg/dx + df/dxgSo: d(8xy)/dx = 8(1 * y + x * dy/dx) = 8y +8 x * dy/dxDifferentiating 5y² we use the chain rule: that is if dg/dx = dg/dy * dy/dxSo: d(5y²)/dx = d(5y²)/dy * dy/dx = 10y * dy/dxAs 4 is a constant the differential of it is 0We are then left with: 6x + 8y +8x(dy/dx )+10y(dy/dx) = 0 We can divide through by 2 here : 3x + 4y +4x(dy/dx) +5y(dy/dx) = 0Rearranging and factorising we arrive at: dy/dx (4x+5y) = -3x -4ySo, dividing through by (4x + 5y) we get dy/dx = -(3x +4y)/(4x+5y)