Question

Find y' if arctan (xy) = 9 + (x^4)y.

Answer 1

http://www.wolframalpha.com/input/?i=arctan+%28xy%29+%3D+9+%2B+%28x%5E4%29y

Choose the second of the implicit derivatives

Answer 2

remember:
\[f(x)=\arctan(u)\]\[f \prime(x)=\frac{u \prime}{1+u ^{2}}\]the problem implicit derivates:\[\arctan(xy)=9+x ^{4}y\]\[\frac{y+xy \prime}{1+x ^{2}y ^{2}}=4x ^{3}y+x ^{4}y \prime\] so factorize y':
\[y \prime=\frac{4x ^{3}y+4x ^{5}y ^{3}-y}{x-x ^{4}-x ^{6}y ^{2}}\]