The Curve, C, has equation: x^2 - 3xy - 4y^2 +64 =0 Find dy/dx in terms of x and y. [Taken from Edexcel C4 2015 Q6a]

The question asks us to find dy/dx, in terms of x and y. Here though it is not in the form of y=f(x), but a mixed form, which we will not be able to separate out. We will have to use Implicit differentiation. Differentiate both sides of this equation and then rearrange it to isolate the dy/dx term on one side. So: d/dx ( x^2 - 3xy -4y^2 + 64 ) = d/dx(0).

Well, we know that to differentiate, x^n, we get n*x^(n-1). So, d/dx( x^2 ) = 2x.

For the next term, we must use the product rule. You can think of this as two functions of x multiplied together. f(x)y(x). The product rule tells us we must differentiate one and then multiple by the other, for each of the terms. So, d/dx( -3xy ) = d/dx(-3x)y + d/dx(y)-3x = -3y -3xdy/dx.

Using Implicit differentiation, we know that to differentiate y, with respect to x, you differentiate y normally, and then you have to multiply it by dy/dx. For the last term, differentiate implicitly again: d/dx(-4y^2)= -8y*dy/dx. And differentiating a constant term, gives 0. So, after differentiating, we get that:

2x - 3y - 3xdy/dx - 8ydy/dx = 0.

Now factorise out the common term, -1*dy/dx, to try and isolate it to one side : 2x - 3y - dy/dx * ( 3x + 8y ) = 0.

Then simply rearrange this to get the dy/dx term on its own, and you get the answer:

dy/dx = (2x - 3y)/(3x + 8y).