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One possibility would be to rearrange the equation to solve for y explicitly and then use standard procedures to find the derivative. However, this is not always possible, and is indeed difficult to do here.
Instead, the procedure of implicit differentiation involves taking the derivative of both sides of the equation with respect to x. Remember that y is a function of x, so taking the derivative of a function of y involves using the chain rule:
| xy2 = x + y |
Differentiate both sides with respect to x First consider xy2. x and y are both functions of x, so the product rule is needed here. Also, the chain rule is needed to evaluate the derivative of y2 with respect to x:
  xy2 | = (1) y2 + x 2y  | ||
| (x + y) | = 1 + 1 |
Recombining both sides of the equation yields:
| (1)y2 + x2y = 1 + 1 |
Now, solving algebraically for 
yields
=  |
Finally, the slope at (0,0) is
=  = - 1 |
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