I am given the equation: $x^{a}y^{b} = 6$
Using implicit differentiation I find that the derivative of the equation with respect to y gives: $\frac{d}{dx}(y) = -\frac{ay}{bx}$. However when I attempt to differentiate the "regular" way I don`t seem to reach the same answer. I would greatly appreciate someone walking me through the problem.
then we get $$ax^{a-1}y^b+x^aby^{b-1}y'=0$$ you must use the product and chain rule: $$(uv)'=u'v+uv'$$ and since $$y=y(x)$$ the first derivative is given by $$y'(x)$$ $$(x^a)'=ax^{a-1}(x)$$ and $$(y(x)^b)'=by(x)^{b-1}\cdot y'(x)$$