Question

Consider the equation:

x^2 - 2xy + 4y^2 = 64

Write an expression of the slope of the curve at any point.

(y^p)= y prime

My work

2x - 2(xy^p + y) + 8yy^p = 0

2x -2xy^p - 2y + 8yy^p = 0

-2xy^p + 8yy^p = 2y -2x

factored out y^p and then divided to get

(2y - 2x) / (8y - 2x)

Which I think is wrong.

Please let me know where I went wrong if I did.

Answer 1

I would reduce it to

y'=(y-x)/(4y-x)

Answer 2

Ok, so it was right at first. Thanks for the further simplifying. :)

Answer 3

[AI-generated answer — Step-by-Step Bot]

To find the expression for the slope of the curve at any point on the given equation, we need to take the derivative with respect to x.

Given equation: x^2 - 2xy + 4y^2 = 64

Taking the derivative of both sides with respect to x, we get:

d/dx (x^2 - 2xy + 4y^2) = d/dx (64)

2x - 2y(dy/dx) + 8y(dy/dx) = 0

Now, let's rearrange the equation to solve for dy/dx:

2x + (8y - 2y)(dy/dx) = 0

2x + 6y(dy/dx) = 0

dy/dx = -2x / (6y)

Simplifying further, we have:

dy/dx = -x / (3y)

So, the expression for the slope of the curve at any point is -x / (3y).

Answer 4

[AI-generated answer — Explain Bot]

To find the expression for the slope of the curve at any point on the given equation, we need to find the derivative of the equation with respect to x and then solve for y'.

Starting with the equation:
x^2 - 2xy + 4y^2 = 64

We can differentiate both sides of the equation with respect to x using the chain rule for the terms containing y:

d/dx(x^2) - d/dx(2xy) + d/dx(4y^2) = d/dx(64)

Using the power rule, the derivatives of the terms are:

2x - (2y + 2xy') + 8yy' = 0

Now, let's rearrange the equation to solve for y':

2x - 2y - 2xy' + 8yy' = 0

Grouping the terms containing y' together:

(-2xy' + 8yy') = (2y - 2x)

Factoring y':

y'(-2x + 8y) = (2y - 2x)

Now, solve for y':

y' = (2y - 2x) / (8y - 2x)

So, the expression for the slope of the curve at any point on the given equation is:

y' = (2y - 2x) / (8y - 2x)

Therefore, your work is correct, and the expression you obtained is the correct expression for the slope of the curve.