Suppose xy=4 and dy/dt=-2. Find dx/dt when x=-3

Dy/dx = dy/dt * dt/dx

xy = 4

y + x (dy/dx) = 0 by implicit differentiation.

x (dy/dx) = - y

dy/dx = - y/x

dy/dx = dy/dt * dt/dx dy/dt = - 2

-y/x = - 2 * dt/dx

y / (2x) = dt/dx

dt/dx = y / (2x)

dx/dt = 2x/y

When x = - 3, xy = 4, y = 4/x = 4/-3 = - 4/3

dx/dt = 2*-3 / (-4/3) = - 6 * - 3/4 = 18/4 = 9/2 = 4.5

dx/dt = 4.5

We are given with the expression xy = 4. The first derivative of the x is dx/dt while that of y is dy/dt. in this case using the multiplication rule of differentiation and that of constants, x dy/dt + y dx/dt = 0. when x = - 3, y should be - 4/3. Hence when dy/dt = - 2, - 3 (-2) - 4/3 * dx/dt = 0. dx/dt is equal to 9/2.