SOLUTION: Find y'' if x^4 + y^4 = 25. Don't solve for dy/dx after taking the first derivative. Write answers in terms of x and y.

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Question 1027550: Find y'' if x^4 + y^4 = 25. Don't solve for dy/dx after taking the first derivative. Write answers in terms of x and y.
Found 2 solutions by Fombitz, robertb:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Differentiate once,
1.4x%5E3%2B4y%5E3y%5B1%5D=0
Differentiate twice,
12x%5E2%2B4%28y%5E3y%5B2%5D%2By%5B1%5D%2A3y%5E2%29=0
where y%5B1%5D is the first derivative, y%5B2%5D is the second.
y%5E3y%5B2%5D=-3%28y%5E2%28y%5B1%5D%29%5E2%2Bx%5E2%29%29
y%5B2%5D=%28-3%28y%5E2%28y%5B1%5D%29%5E2%2Bx%5E2%29%29%2Fy%5E3%29
Then from (1),
y%5B1%5D=-x%5E3%2Fy%5E3
y%5B1%5D%5E2=x%5E6%2Fy%5E6
Substituting,
y%5B2%5D=%28-3%28y%5E2%28x%5E6%2Fy%5E6%29%2Bx%5E2%29%29%2Fy%5E3%29
y%5B2%5D=%28-3%28%28x%5E6%2Fy%5E4%29%2Bx%5E2%29%29%2Fy%5E3%29
y%5B2%5D=%28-3x%5E2%28%28x%5E4%2Fy%5E4%29%2B1%29%29%2Fy%5E3%29
y%5B2%5D=%28-3x%5E2%28x%5E4%2By%5E4%29%29%2Fy%5E7%29

Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Use implicit differentiation.
==> 4x%5E3%2B+4y%5E3%2A%28dy%2Fdx%29+=+0
==> x%5E3%2B+y%5E3%2A%28dy%2Fdx%29+=+0 ==> -x%5E3%2Fy%5E3=dy%2Fdx
Differentiate again:
3x%5E2%2B3y%5E2%28dy%2Fdx%29%28dy%2Fdx%29+%2B+y%5E3%28d%5E2y%2Fdx%5E2%29+=+0
==> -3x%5E2-3y%5E2%28dy%2Fdx%29%5E2+=+y%5E3%28d%5E2y%2Fdx%5E2%29
==> %28-3x%5E2%29%2Fy%5E3-%283%28dy%2Fdx%29%5E2%29%2Fy+=+d%5E2y%2Fdx%5E2
==> %28-3x%5E2%29%2Fy%5E3-%283%28-x%5E3%2Fy%5E3%29%5E2%29%2Fy+=+d%5E2y%2Fdx%5E2
==> %28-3x%5E2%28y%5E4%2Bx%5E4%29%29%2Fy%5E7+=+d%5E2y%2Fdx%5E2
==> %28-3x%5E2%2A25%29%2Fy%5E7+=+d%5E2y%2Fdx%5E2
==> %28-75x%5E2%29%2Fy%5E7+=+d%5E2y%2Fdx%5E2