Related Rates Calculus Question

Question

if y^2 + xy - 3x = 14, and dy/dt = -4 when x = -2 and y = -2, what is dx/dy when x= -2 and y= -2

Answer 1

dx/dy = (dx/dt)/(dy/dt)

dy/dt is given as -4 when (x,y) = (-2,-2)

Need to find dx/dt from the equation using implicit derivation.

y²+xy-3x=14

Taking the derivative of both sides:

2ydy/dt + xdy/dt+ydx/dt - 3dx/dt =0

(2y+x)dy/dt = (3-y)dx/dt

dx/dt = ((2y+x)/3-y))dy/dt

at (-2,-2)

dx/dt = (-6/5)(-4)=24/5

dx/dy = (24/5)/-4 = -6/5

Answer 2

y^2 + xy - 3x = 14

2yy' + xy' + yx' - 3x' = 0

2(-2)(-4) + (-2)(-4) + (-2)x' -3x' = 0

16 + 8 -(2+3)x' = 0

-5x' = -24

x' =-24/-5 = 24/5 =4 4/5

dx/dt = 24/5 when dy/dt =-4 and x=-2, y=-2

dy/dx = dx/dt over dy/dt = -4/24/5 = -5/6

dy/dx = -5/6

flip the fraction upside down to get

dx/dy =-6/5

or just take the derivative of y with respect to x in the original equation y^2 + xy -3x = 14

y' = dy/dx = 2yy' +xy' + y -3 = 0

y'(-2) = 2(-2)y' + (-2)y' + (-2) - 3 = 0

-4y' -2y' -2 -3 =0

(-4-2)y' = 5

y'= -5/6

dy/dx = -5/6

dx/dy = (1/dy/dx) = -6/5