Curve C has equation x^2 - 3xy - 4y^2 + 64 = 0. a) find dy/dx in terms of x and y. b) find coordinates where dy/dx=0.

a) This is no standard differentiate question. You will need identify the fact that the equation contains both x and y terms and thus you will need to differentiate with respect to both. This involves applying both the chain rule. and the product rule. Breaking it down into terms. the first term goes to 2x through normal differentiation. The second term requires the product rule whereby dy/dx= u.dv/dx + v.du/dx) to find -3y - 3x.dy/dx. The Third term applies the chain rule (dy/dx=dx/dy . dy/dx) and perform implicit differentiation to find -8y.dy/dx and finally the last term goes to 0. Then we will need to simplify and rearrange to find dy/dx = (2x-3y)/(-3x-8y).
b) It is important to first recognise the nature of the problem. The denominator cannot equal 0. Therefore 2x-3y=0. Once we rearrange the equation in terms of either x or y we can substitute it back into the original equation. This allows us to find a coordinate for x or y. Then using 2x-3y=0 we can use the solution found to find the other variable. It is important to note that when finding a value of x from a squared x such as x^2=a number, means that both the positive and negative values must be taken. Therefore the solutions are (24/5 , 16/5) and (-24/5 , -16/5).