find dy/dt and dx/dt

Question

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

x2 + y2 = 169

(a) Find dy/dt, given x = 5, y = 12, and dx/dt = 8.

dy/dt =

(b) Find dx/dt, given x = 12, y = 5, and dy/dt = –4.

dx/dt =

Answer 1

Hi Artavius,

The biggest part is finding the expression for dy:

1) Express for y: y = (169 - x²)^1/2

2) Use the chain rule: d/dx(f(g(x)) = f'(g(x))g'(x), where g(x) = 169 - x²

dy = -x(169 - x²)^-1/2dx. For convenience, we can even substitute y=(169 - x²)^1/2 back in to get

dy = -(x/y)dx

3) we get the expression we need for (a) by dividing both sides by dt:

dy/dt = -(x/y)dx/dt

4) so for x = 5, y = 12, and dx/dt = 8,

dy/dt = -(5/12)8 = -10/3

5) for part (b), just express step (3) in terms of dx/dt:

dx/dt = -(y/x)dy/dt

6) so for x = 12, y = 5, and dy/dt = –4,

dx/dt = -(5/12)(-4) = 5/3