I am having trouble differentiating x^2 + 2lny = y (solving for dy/dx). I know that I need to do implicit differentiation, but the answer I get is not correct. The answer I got was (2x-2xy)/2, but the answer is (2xy)/(y-2). How do I do this problem?
x^2 + 2lny = y
2x + 2/y (dy/dx) = dy/dx
times y
2xy + 2 dy/dx = y dy/dx
2xy = y dy/dx - 2 dy/dx
2xy = dy/dx (y - 2)
dy/dx = 2xy/(y-2)
Thank you! Now I understand that I should've multiplied by y instead of moving the whole "2/y (dy/dx)" to the other side of the equation and then factor the dy/dx out. I can't believe how such a simple manuever can make the problem so much easier to solve. Thanks again!
To solve for dy/dx in the equation x^2 + 2lny = y, you're right to use implicit differentiation. However, it seems like there might be an error in your calculation. Let's go through the process step by step to find where the mistake occurred.
Step 1: Start by differentiating both sides of the equation with respect to x. This involves applying the chain rule whenever necessary.
For the left side:
d/dx (x^2) = 2x
For the right side:
To differentiate y, we can use the chain rule. Let u = y, so du/dx = dy/dx.
d/dx (lnu) = (1/u) * du/dx = (1/y) * dy/dx
Step 2: Combine the derivatives from each side of the equation.
2x + (1/y) * dy/dx = dy/dx
Step 3: Isolate dy/dx.
To do this, subtract dy/dx from both sides of the equation.
2x + (1/y) * dy/dx - dy/dx = 0
Simplify the equation.
2x + (1/y - 1) * dy/dx = 0
Now, multiply through by y to get rid of the fraction.
(2xy) + (1 - y) * dy/dx = 0
Step 4: Solve for dy/dx.
Subtract (2xy) from both sides.
(1 - y) * dy/dx = -2xy
Finally, divide both sides by (1 - y).
dy/dx = -2xy / (1 - y)
So the correct answer for dy/dx is (-2xy) / (1 - y).