Question

Assume x=x(t) and y=y(t). Find dx/dt if x^2(y-6) = 12y + 3 and dy/dt = 2 when x = 5 and y = 12

Answer 1

$x^2(y-6)=12y+3$

Differentiate:

$2x\dfrac{\mathrm dx}{\mathrm dt}(y-6)+x^2\dfrac{\mathrm dy}{\mathrm dt}=12\dfrac{\mathrm dy}{\mathrm dt}$

Plug in everything you know: $\dfrac{\mathrm dy}{\mathrm dt}=2$, $x=5$, and $y=12$

$2(5)\dfrac{\mathrm dx}{\mathrm dt}(12-6)+5^2(2)=12(2)\implies\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac{13}{30}$