Question

Suppose that x=x(t) and y=y(t) are both functions of t. If x^2+xy=8, and dx/dt=−5 when x=2 and y=2, what is dy/dt?

Answer 1

2x dx/dt+y dx/dt+x dy/dt=0. Since x=2, y=2, and dx/dt=-5, dy/dt=15.

Answer 2

$x^2+xy=8$

Differentiating both sides with respect to $x$ yields

$2x+x\dfrac{\mathrm dy}{\mathrm dx}+y=0\implies\dfrac{\mathrm dy}{\mathrm dx}=-\dfrac{2x+y}x$

Now,

$\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}$

So you have

$-\dfrac{2x+y}x=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}$

When $x=2$ and $y=2$, you have $\dfrac{\mathrm dx}{\mathrm dt}=-5$, which means

$-\dfrac{2(2)+2}2=-\dfrac{\frac{\mathrm dy}{\mathrm dt}}5$
$\dfrac{\mathrm dy}{\mathrm dt}=15$