Suppose that y is a function of x determined by e^xy = y. Find y(0) and y'(0).
I'm assuming this is $\displaystyle e^{xy} = y$ and not $\displaystyle e^xy = y$ (which would be kind of silly, I guess)?
y(0):
$\displaystyle e^{0 \cdot y} = y$
$\displaystyle y = 1$
So y(0) = 1.
y'(0):
$\displaystyle e^{xy} = y$
$\displaystyle (y + xy')e^{xy} = y'$
So at x = 0 this becomes:
$\displaystyle (y(0) + 0 \cdot y')e^{0 \cdot y} = y'(0)$
$\displaystyle y(0) \cdot 1 = y'(0)$
Since y(0) = 1, thus y'(0) = 1 also.
-Dan