12th Grade Calculus
posted by Liz on .
Find d^2y/dx^2 by implicit differentiation.
x^(1/3) + y^(1/3) = 4
I know that first you must find the 1st derivative & for y prime I got 1/3x^(2/3) + 1/3y^(2/3) dy/dx = 0
Then for dy/dx I got
dy/dx = [1/3x^(2/3)] / [1/3y^(2/3)]
I think that from here I would use the quotient rule to find the second derivative?

simplify your first derivative before going further
notice you can divide each term by 1/3 to get
x^(2/3) + y^(2/3) dy/dx = 0
dy/dx = x^(2/3) / y^(2/3)
=  y^(2/3)/x^(2/3)
=  (y/x)^(2/3)
d^2y/dx^2 = (2/3) (y/x)^(1/3) [( xdy/x  y)/x^2)
replace the dy/dx in the square bracket by (y/x)^2/3) and see what you get.
Still messy but a small improvement. 
ggggggggggggggggggggg