Question

assume that x and y are both differentiable functions of t and find rhe required values of dy/dt and dx/dt. xy=4. find dy/dt, given x=2 and dx/dt =10

Answer 1

Differentiate both sides with respect to $t$, using the product and chain rules on the left side.

$xy = 4 \implies x\dfrac{dy}{dt} + y \dfrac{dx}{dt} = 0$

When $x=2$, the given equation tells us that $2y=4\implies y=2$. Then if $\frac{dx}{dt}=10$, it follows that

$2\dfrac{dy}{dt} + 2\times10 = 0 \implies \dfrac{dy}{dt} = \boxed{-10}$