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.