Implicit Differentiation to find dy/dx

[akstradtne]

Use implicit differentiation to find dy/dx given x^2y+xy^2=4.

I have no idea how to approach this problem. My instructor assigned this as homework but has not gone over it at all in class. We have gone over explicit differentiation and I understand this well. I have read the section but it is making no sense. I think I need to use the General Power Rule: d/dx(y^n)=ny^(n-1)dy/dx but I don't know how to use it.

 

[Dick]

Use implicit differentiation to find dy/dx given x^2y+xy^2=4.

I have no idea how to approach this problem. My instructor assigned this as homework but has not gone over it at all in class. We have gone over explicit differentiation and I understand this well. I have read the section but it is making no sense. I think I need to use the General Power Rule: d/dx(y^n)=ny^(n-1)dy/dx but I don't know how to use it.

You've got all the ingredients. Differentiate both sides with respect to x and solve for dy/dx. Use the product rule on the way.

 

[akstradtne]

Can you explain a little more. I don't understand "with respect to x or y." How do I use the general power rule?

 

[HallsofIvy]

You said, with reference to implicit differentiation, "I understand this well". Did you mean to say "I don't understand this well"?

Differentiating "with respect to x" means treating y as a function of x, not a separate variable. So "$y^n$, differentiated "with respect to y" would be just $d(y^n)/dy= ny^{n-1}$. But differentiating "with respect to x" you would use the chain rule: $$d(y^n)/dx= ny^{n-1} (dy/dx)$$.

If you had, for example, $x^2y+ xy^2= x- y$, differentiating each part with respect to x, (using the "product rule" for the terms on the left), $2xy+ x^2(dy/dx)+ (1)y^2+ 2xy(dy/dx)= 1- dy/dx$. You can then solve that equation for dy/dx.