WAEC 2020 · Paper 2 · Q1✱

If $x^2 - xy - y^2 = -31$ , find $\dfrac{dy}{dx}$ at $(2, 5)$ .

Plot the curves, move them, and read values off the graph.

Worked solution (try it first)

Differentiate each term with respect to $x$ .
$x^2$ gives $2x$ .
$-xy$ needs the product rule and gives $-\left(y + x\dfrac{dy}{dx}\right)$ .
$-y^2$ gives $-2y\dfrac{dy}{dx}$ .
$-31$ gives 0.
So $2x - y - x\dfrac{dy}{dx} - 2y\dfrac{dy}{dx} = 0$ .
Collect: $2x - y = (x + 2y)\dfrac{dy}{dx}$ , so $\dfrac{dy}{dx} = \dfrac{2x - y}{x + 2y}$ .
At $(2, 5)$ : $\dfrac{dy}{dx} = \dfrac{4 - 5}{2 + 10}$
$= -\dfrac{1}{12}$ .