Implicit Differentiation... the triple product rule? When implicitly finding the derivative of:  

$xy^3 - xy^3\sin(x) = 1$

How do you find the implicit derivative of:

$xy^3\sin(x)$

Is it using a triple product rule of sorts?

HINT $\;$ logarithmic differentiation makes the n-ary generalization obvious:
$$\rm (abc)'= abc \;  log(abc)' = abc \;(log\; a + log\; b + log\; c)' = abc \; \bigg(\frac{a'}{a} + \frac{b'}{b} + \frac{c'}{c}\bigg) $$
Obviously the same proof works for arbitrary length products yielding
$$\rm (abc\cdots f)' = \: abc\cdots f\;\bigg(\frac{a'}{a} + \frac{b'}{b} + \frac{c'}{c} +\:\cdots\:+ \frac{f'}{f}\bigg) $$
