solve for dy/dx when x = (x/y) - 4y

Question

I solved the equation by getting y on one side and then solve for the derivative using the quotient rule. I got:

dy/dx = [4y - 4x(dy/dx)] / [(x+4y)^2]

Is it ok if I leave dy/dx in my solution? How would you go about solving this? Or is there a better way to go about solving it?

Thank you for your time.

Answer 1

Differentiate both sides to get

Multiply both sides by y² to get

y² = y – xy' – 4y²y'

Now add xy' + 4y²y' to, and subtract y² from, both sides:

xy' + 4y²y' = y – y²

Factor the left hand side:

y'(x + 4y²) = y – y²

Divide both sides by x + 4y² to get your answer:

Answer 2

Usually, you want to isolate the dy/dx term when doing implicit derivation. From the input given, I think

you made a mistake in doing the implicit derivation for the first step.

x = (x/y)-4y

Doing the implicit derivation:

1 = (y-xy')/y²-4y'

y² = y-xy' -4y²y'

y²-y = -y'(x+4y²)

y' = dy/dx = (y-y²)/(x+4y²)