What is the derivative of # (x^3)-(xy)+(y^3)=1#?

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Answer 1 · 2017-08-16T17:01:11

#dy/dx=(y-3x^2)/(3y^2-x)#

We need to find the derivative with respect to #x#, #dy//dx#.

When we differentiate #y#, since it's a function that's not #x#, the chain rule will kick in and a #dy//dx# term will arise thanks to the chain rule.

Also don't forget that differentiating #xy# will use the product rule.

Differentiating:

#d/dx(x^3-xy+y^3)=d/dx(1)#

#3x^2-(d/dxx)y-x(d/dxy)+3y^2(d/dxy)=0#

#3x^2-y-xdy/dx+3y^2dy/dx=0#

Solve for #dy/dx#:

#dy/dx(3y^2-x)=y-3x^2#

#dy/dx=(y-3x^2)/(3y^2-x)#