Find dy/dx
2x^2 y^2 = x^3 y^3
I got dy/dx = (3x^2 y^3 - 4xy^2) / (4x^2 y - 3x^3 y ^2) but the answer is apparently different. Can anyone help me with this question?
2x^2 y^2 = x^3 y^3
I would first simply it to
2 = xy , where x, y ≠ 0
0 = x dy/dx + y
dy/dx = -y/x
use
www.desmos.com/calculator
to graph both xy = 2 and
2x^2 y^2 = x^3 y^3
on the same grid to show that they are the same
2x^2 y^2 = x^3 y^3
4xy^2 + 4x^2 yy' = 3x^2 y^3 + 3x^3 y^2 y'
y'(4x^2 y - 3x^3 y^2) = 3x^2 y^3 - 4xy^2
as you said. But maybe you should have started by dividing both sides by x^2 y^2 to get
xy = 2
y + xy' = 0
y' = -y/x
I'm sure that with enough algebra, you could manipulate your answer into this form -- give it a shot
To find dy/dx, we can use implicit differentiation. Here's how you can go about it step by step:
1. Start by differentiating both sides of the equation with respect to x. Remember to apply the product rule when differentiating terms that involve y.
2x^2 y^2 = x^3 y^3
Differentiating both sides with respect to x:
d/dx (2x^2 y^2) = d/dx (x^3 y^3)
Now, let's differentiate each term separately:
d/dx (2x^2 y^2) = d/dx (2x^2) * y^2 + 2x^2 * d/dx (y^2)
= 4x * y^2 + 2x^2 * 2y * dy/dx
d/dx (x^3 y^3) = d/dx (x^3) * y^3 + x^3 * d/dx (y^3)
= 3x^2 * y^3 + x^3 * 3y^2 * dy/dx
2. Simplify the resulting equation:
4xy^2 + 4x^2y(dy/dx) = 3x^2y^3 + 3xy^2(x^2)(dy/dx)
3. Move all terms containing dy/dx to one side of the equation:
4x^2y(dy/dx) - 3xy^2(x^2)(dy/dx) = 3x^2y^3 - 4xy^2
Factor out the dy/dx:
dy/dx(4x^2y - 3xy^2(x^2)) = 3x^2y^3 - 4xy^2
4. Solve for dy/dx by dividing both sides by (4x^2y - 3xy^2(x^2)):
dy/dx = (3x^2y^3 - 4xy^2) / (4x^2y - 3xy^2(x^2))
Your answer is correct! The calculated derivative matches the expression you provided. If there is a different answer mentioned, it might be helpful to double-check the specific interpretation or context where the different answer is stated.