# Help regarding implicit differentiation.

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.

• is here assumed to be $$y=y(x)?$$ – Dr. Sonnhard Graubner Oct 3 '17 at 20:05
• $x^ay^b$ is an expression not an equation – Riley Oct 3 '17 at 20:05
• You are right, I edited it now. Yes, y =y(x) – user487324 Oct 3 '17 at 20:09
• now it is an equation – Dr. Sonnhard Graubner Oct 3 '17 at 20:09

## 1 Answer

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 $$(y(x)^b)'=by(x)^{b-1}\cdot y'(x)$$

• Could you differentiate it regularly for me? Preferably step-by-step. – user487324 Oct 3 '17 at 20:13
• Thank you for the help! – user487324 Oct 3 '17 at 20:21
• it is nice that i could help you! all the best for you! – Dr. Sonnhard Graubner Oct 3 '17 at 20:23