SOLUTION: If y=x²+y²=2y, find dy/dx in terms of x and y. Prove that d²y/dx²=1/(1-y)³

Algebra.Com
Question 1204134: If y=x²+y²=2y, find dy/dx in terms of x and y. Prove that d²y/dx²=1/(1-y)³
Found 3 solutions by ikleyn, Edwin McCravy, math_tutor2020:
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
.

As equation is written in the post with two equation signs,

it contradicts to basic elementary Math grammar rules and is not readable.



Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
I'm sorry but this
 
y=x²+y²=2y

makes no sense unless x and y were both equal to zero.

y just doesn't equal to 2y any other time. That is not what you meant.

please tell me what you meant in the space provided below for a thank you
note, and I'll help you.

Edwin
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Through a bit of trial-and-error, it appears the given equation should be .
Please be more careful when entering the equation.

The 1st derivative would be:
The scratch work is shown in the next section.

------------------------------------



















This will be used later.

------------------------------------

We then apply another round of implicit differentiation with respect to x.





Quotient Rule







From here we plug in the first derivative
















Use the equation x^2+y^2 = 2y




------------------------------------

Here's another way to do it





Product Rule





Plug in dy/dx = x/(1-y)













Replace x^2+y^2 with 2y




RELATED QUESTIONS

If y=x/x+1,dy/dx=? (answered by ikleyn,greenestamps)
Given that dy/dx=a/x^2+1 and that when x=1 dy/dx=3 and y=3 find the value of y when... (answered by solver91311)
Given that dy/dx=a/x^2 +1 and that when x=1 dy/dx=3 and y=3 find the value of y when... (answered by solver91311)
Use parametric differentiation to find dy/dx in terms of t, then evaluate dy/dx when t = (answered by ikleyn)
I have a question regarding finding the second derivative of a hyperbola... Find the... (answered by Alan3354)
Dy/dx=(x+y+1)^2 (answered by MathLover1)
y= 1/∛x fine... (answered by Boreal)
if y=x+(a^2+x^2)^1/2 and a is a constant prove that... (answered by solver91311)
if y ^(y) = e ^(x - y) find (dy)/(dx) at x =... (answered by math_tutor2020)