Answer:

Explanation:
To find dy/dt given dx/dt, we need to first find dy/dx, since:

To find dy/dx, differentiate the given equation that contains a mixture of x and y terms with respect to x using implicit differentiation.
Begin by placing d/dx in front of each term of the equation:

Differentiate the terms in x only (and constant terms):

Use the chain rule to differentiate terms in y only.
In practice, this means differentiate with respect to y, and place dy/dx at the end:

Use the product rule to differentiate the term in x and y.

Rearrange the resulting equation to isolate dy/dx:




Now we have found dy/dx, and given that dx/dt = -8, we can create an expression for dy/dt:




To find the value of dy/dt when x = 4 and y = -1, substitute these values into the found expression for dy/dt:




Therefore, the value of dy/dt for the given conditions is -7/3.