Partial derivatives question help here?

Question

I have the function $$u=\frac{x^3+8xy^2-y^3}{x^3+y^3}.$$ I have to show if $$x\cdot \left(\frac{\partial u}{\partial x}\right) +\left(\frac{\partial u}{\partial y} \right)=0.$$

How do I find du/dx and du/dy?

Answer 1

You calculate $$\frac{\partial u}{\partial x}=\frac{2 y^2 \left(-8 x^3+3 x^2 y+4 y^3\right)}{\left(x^3+y^3\right)^2}$$ by taking $y$ constant and just differentiate a function with one variable.

The partial derivative in respect to $x$ of the nominator is $$ \frac{\partial }{\partial x} (x^3+8xy^2-y^3)=3x^2 + 8y^2$$ and the partial derivative in respect to $x$ of the denominator is $$\frac{\partial }{\partial x} (x^3+y^3)= 3 x^2$$