# Differentiating with respect to x with y in the equation

find $\dfrac{\mathrm d y}{\mathrm dx}$ of $x^2y+xy^2$

I am unsure on what to do about the $y$ in the expression

Thanks.

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You can't compute rate of change of one variable w.r.t. other variable from an expression, it has to be an equation that relates those variables(or implies that they are independent) –  Aang Mar 9 '13 at 6:17
Is the equation equal to some constant? –  Inceptio Mar 9 '13 at 6:25
The expression - the whole problem is that it is not an equation :) @Inceptio –  Thomas Andrews Mar 9 '13 at 6:26
If the question asked you explicitly to find $\frac{dy}{dx}$, then you were probably told that $x^2y+xy^2=17$, or $x^2y+xy^2=x^{17}$, or something like that. The expression "find $\frac{dy}{dx}$ of $x^2y+xy^2$" doesn't make sense, so is unlikely to be what you were asked to do. –  André Nicolas Mar 9 '13 at 6:41
Then unfortunately, before you can do the math problem, you must first solve the riddle "What question did your book mean to ask?" –  Hurkyl Mar 10 '13 at 14:01

If you had to find $dy/dx$, where, for example, $x^2y+xy^2=7.$ Then you could take the derivative of both sides with respect to $x$:

$\frac d{dx}(x^2y+xy^2)=\frac d{dx}7.$

This means that $\frac d{dx}(x^2y+xy^2)=0.$

Now, since you are interested in changes in $x$ you treat $y$ as an unknown function of $x$ and use the chain rule (and in this case the product rule):

$2xy + (dy/dx)x^2+y^2+2xy(dy/dx)=0.$

So that $(dy/dx)(x^2+2xy)=-2xy-y^2$ and so $dy/dx=(-2xy-y^2)/(x^2+2xy).$

This is called implicit differentiation. It is just the chain rule applied to an implied function of $x$ but, as has been mentioned in the comments, we need an equation to make it an implied function of $x$.

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