Suppose that x=x(t) and y=y(t) are both functions of t. if x^2+xy=8, and dx/dt=−5 when x=2 and y=2, what is dy/dt?

The differentiation(dy/dt) of y with respect to t of the given function with all the given parameters gives us;

dy/dt = 15

We are given the function;

x² + xy = 8

Differentiating with respect to x gives;

2x + x(dy/dx) + y = 0

Making dy/dx the subject gives;

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

Now, we are told that x = x(t) and y = y(t) are both functions of t. This means that;

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

-(y + 2x)/x = (dy/dt)/-5

At x = 2 and y = 2, we have;

-(2 + 2(2))/2 = (dy/dt)/-5

-3 = (dy/dt)/-5

dy/dt = -3 * -5

dy/dt = 15

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