<p>Can't figure this out, can I get some help?</p>
<p>y^3 + 3yx^2 + 13 = 0</p>
<p>Find dy/dx</p>
<p>Alright. I'll show everything, but some of the steps here are pretty redundant and could be left out.</p>
<p>First, differentiate both sides. So: d/dx(y^3+3yx^2+13)=d/dx(0)
Thus d/dx(y^3)+3<em>d/dx(yx^2)+d/dx(13)=d/dx(0)
Now using the fact that the derivative of a constant term is 0, and the product rule, we get: d/dx(y^3)+3(y</em>d/dx(x^2)+x^2<em>d/dx(y))=0. Then it's just a matter of taking the appropriate derivatives, and using algebraic techinques to solve for dy/dx.
3y^2</em>dy/dx+3(2xy+x^2<em>dy/dx)=0
3y^2</em>dy/dx+6xy+3x^2<em>dy/dx=0
dy/dx</em>(3x^2+3y^2)=-6xy
dy/dx=-6xy/(3x^2+3y^2)
dy/dx=-2xy/(x^2+y^2)</p>
<p>Since it's pretty difficult (assuming it's even possible, which i doubt but didn't actually try) to solve for y in the original expression, i assume that it's ok to leave the derivative defined implicitly for your final answer.</p>
<p>Unless you know the cubic equation or can do some crazy factoring (i'm guessing you can't), implicit diff. is the way to go.</p>
<p>according to mathematica, there is an explicit solution for y... but it is excruciatingly complicated</p>
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