what is the answer?

Question

Assume that x and y are both differentiable functions of t and find the required values of dy/dt and dx/dt.

xy = 6

(a) Find dy/dt, given x = 8 and dx/dt = 11.

dy/dt =  

(b) Find dx/dt, given x = 1 and dy/dt = –5.

dx/dt = 

Answer 1

xy = 6, x=8, x'= 11

y= 6/x = 6x^-1

y' = (-6/x^2)x'

y'(8) = (-6/64)(11) = -66/64 = -33/32

y' dy/dt= -33/32

y'5 x=1, xy=6

x= 6/y

x' = (-6/y^2)(y') =(-6/(6/x)^2(-5) = (-6/(36/x^2)(-5) = 30x^2/36 = 30/36 = 15/18 = 5/6

x' = 5/6

dx/dt = 5/6

Answer 2

xy = 6; d/dt(xy) = 0; ydx/dt + xdy/dt = 0.

(a) x = 8; 8y = 6; y = 3/4. So, 3/4·11 + 8dy/dt = 0; dy/dt = - 33/32.

(b) x = 1, then y = 6; 6dx/dt - 1·5 = 0; dx/dt = 5/6