Question

If x^2=y^2+2y+6 and dx/dt=8. Find dy/dt at the point (3,1). I don't know how to find it at the point or how to start finding the derivatives.

Answer 1

To find dy/dt at the point (3,1), differentiate the equation x^2 = y^2 + 2y + 6 to find dy/dx. Then, substitute the values of x = 3 and y = 1 into the derivative to find dy/dx at the given point: dy/dt = 6.

Step-by-step explanation:

To find dy/dt at the point (3,1), we need to find the derivative of y with respect to t and then substitute the values of x and t at the given point into the derivative.

First, let's find the derivative of y. Differentiating both sides of the equation x^2 = y^2 + 2y + 6, we get 2x = 2yy' + 2y', where y' represents dy/dx. Rearranging, we have 2y' = 2x - 2yy'. Solving for y', we get y' = (2x) / (2 - 2y).

Now, substitute the values of x = 3 and y = 1 into the derivative to find the value of dy/dx at the given point: dy/dt = (2*3) / (2 - 2(1)) = 6. Therefore, dy/dx = 6 at the point (3,1).

Answer 2

dx/dt = ±3 when y = 6