Find dy/dx by implicit differentiation:
x^2-2xy+y^3 = c
Thank you!
Well, well, well, it seems we have an implicit differentiation problem on our hands. Let's get to it, shall we?
First, let's give ourselves a friendly reminder that the derivative of a constant c with respect to x is simply 0. Now, let's get cracking.
Taking the derivative of both sides with respect to x, we get:
(2x - 2y * dy/dx) - (2y * dy/dx) + 3y^2 * dy/dx^3 = 0
Rearranging things a bit, we find:
dy/dx * ( -2y - 2y + 3y^2 * dy/dx^2) = -2x
Simplifying that expression, we have:
dy/dx * (3y^2 * dy/dx^2 - 4y) = -2x
Finally, we can solve for dy/dx:
dy/dx = -2x / (3y^2 * dy/dx^2 - 4y)
Now, I must warn you, this expression may be a bit chaotic. But hey, chaos can be fun, right? Just proceed with caution and you'll be on your way to finding dy/dx in no time.
Good luck, my friend!
To find dy/dx by implicit differentiation, we need to differentiate both sides of the equation with respect to x.
Differentiating the left side of the equation term by term, we get:
d/dx(x^2) - d/dx(2xy) + d/dx(y^3) = d/dx(c)
Using the power rule, we can find the derivatives of the individual terms as follows:
d/dx(x^2) = 2x
d/dx(2xy) = 2y + 2x * (dy/dx) [Using the product rule]
d/dx(y^3) = 3y^2 * (dy/dx) [Using the chain rule]
d/dx(c) = 0 (since c is a constant)
Now, substituting these derivatives back into the equation, we have:
2x - (2y + 2x * (dy/dx)) + 3y^2 * (dy/dx) = 0
Simplifying, we get:
2x - 2y - 2x * (dy/dx) + 3y^2 * (dy/dx) = 0
Rearranging the terms, we obtain:
-2y + (3y^2) * (dy/dx) - 2x * (dy/dx) + 2x = 0
Factoring out (dy/dx), we have:
(3y^2 - 2x) * (dy/dx) = 2y - 2x
Finally, solving for (dy/dx), we find:
(dy/dx) = (2y - 2x) / (3y^2 - 2x)
Therefore, the derivative of y with respect to x, dy/dx, is given by (2y - 2x) / (3y^2 - 2x).
To find dy/dx by implicit differentiation, you can follow these steps:
1. Differentiate both sides of the equation with respect to x.
2. Treat y as a function of x, so y becomes y(x).
3. Use the chain rule to differentiate terms that include y(x).
Let's go through the steps:
1. Differentiate both sides of the equation with respect to x:
d/dx (x^2 - 2xy + y^3) = d/dx (c)
2. Treat y as a function of x, so y becomes y(x):
d/dx (x^2 - 2xy(x) + [y(x)]^3) = 0
3. Use the chain rule to differentiate terms including y(x):
The derivative of x^2 is simply 2x.
To differentiate -2xy(x), we use the product rule:
-2x * y(x) + -2 * y(x) * dy(x)/dx (notice that we used the chain rule here: dy(x)/dx).
To differentiate [y(x)]^3, we use the chain rule again:
3 * [y(x)]^2 * dy(x)/dx.
So the fully differentiated equation becomes:
2x - 2xy(x) - 2y(x) * dy(x)/dx + 3[y(x)]^2 * dy(x)/dx = 0
Now, we can solve for dy/dx, which represents the derivative of y with respect to x.
Rearranging the equation, we get:
dy(x)/dx * (3[y(x)]^2 - 2y(x)) = 2x - 2xy(x)
dy(x)/dx = (2x - 2xy(x))/(3[y(x)]^2 - 2y(x))
And that's the derivative, dy/dx, in terms of x and y.
x² - 2xy + y³ = c
2x - 2y - 2xy' + 3y² y' = 0
2(x-y) + y'(3y²-2x) = 0
y' = 2(y-x)/(3y²-2x)