Calc Problems?

Question

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1. If x^2 + y^2 = 25, what is the value of d^2y/dx^2 at the point (4, 3)?

2. f(x) = ln|x^2-1|, then f' (x) = ?

Answer 1

1. you just do the derivative of each term and when you do a derivative with something with a 'y' in it you add dy/dx to the end and then you solve for dy/dx:
x2 + y2 = 25
2x + 2y dy/dx = 0
dy/dx= -2x/ 2y
then do the quotien rule to find the second derivative.
d2y/dx2 = [ (2y) (2)- (2) (2x) ] / [ (2y)^2 ]

2. the ln rule is divide the ln of whatever comes next by 1 and then multiply it by what you are taking the ln of:
f'(x) = { 1/ [ |x2-1| ] } { 2x}

Answer 2

2x +2yy'=0
y' = -x/y
y" = (-y+xy')/y^2 = (-y -x^2/y)/y^2
y" at (4,3) = (-3 -16/3)/9 = -25/27

f '(x) = 2x/|x^2-1|