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.
I would reduce it to
y'=(y-x)/(4y-x)
Ok, so it was right at first. Thanks for the further simplifying. :)
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).
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.