Given the equation: $$x^2 + y^2 + 10x + 2y - 4xy = 10$$ (a) Find $$\frac{dy}{dx}$$ in terms of x and y, fully simplifying your answer - Edexcel - A-Level Maths Pure - Question 5 - 2014 - Paper 8

To find $\frac{dy}{dx}$, we need to differentiate the given equation implicitly. Starting with:

$x^2 + y^2 + 10x + 2y - 4xy = 10$

Differentiating both sides with respect to x gives:

$2x + 2y\frac{dy}{dx} + 10 + 2\frac{dy}{dx} - 4\left(y + x\frac{dy}{dx}\right) = 0$

Now, rearranging terms leads to:

$2x + 10 + (2 - 4x)\frac{dy}{dx} - 4y = 0$

Reorganizing this, we isolate $\frac{dy}{dx}$:

$(2 - 4x)\frac{dy}{dx} = 4y - 2x - 10$

Thus,

$\frac{dy}{dx} = \frac{4y - 2x - 10}{2 - 4x}$

This is the required expression for $\frac{dy}{dx}$ in terms of x and y.