SOLUTION: if x^2 -y^2=5 what is d^2y/dx^2 at the point (3,2)

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Question 389719: if x^2 -y^2=5 what is d^2y/dx^2 at the point (3,2)
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E2+-+y%5E2+=+5

Differentiating with respect to x,

2x+-+2y%28dy%2Fdx%29+=+0 Isolate dy/dx

dy%2Fdx+=+-2x%2F-2y+=+x%2Fy

Take the second derivative d%5E2y%2Fdx%5E2 using the Quotient rule:

d%5E2y%2Fdx%5E2+=+%28y+-+x%28dy%2Fdx%29%29%2F%28y%5E2%29

Replace dy%2Fdx with x%2Fy

d%5E2y%2Fdx%5E2+=+%28y+-+x%5E2%2Fy%29%2Fy%5E2+=+1%2Fy+-+x%5E2%2Fy%5E3 (splitting up the fraction and simplifying)

Replacing x = 3 and y = 2, d%5E2y%2Fdx%5E2 at the point (3,2) is 1%2F2+-+9%2F8+=+-5%2F8