Answer
$$y'=-\frac{1}{3(xy)^{1/5}}$$
Work Step by Step
$$5x^{4/5}+10y^{6/5}=15$$
$$x^{4/5}+2y^{6/5}=3$$
Using implicit differentiation, we differentiate both sides of the equation with respect to $x$: $$\frac{4}{5}x^{-1/5}+2\times\frac{6}{5}y^{1/5}\times y'=0$$
$$\frac{4}{5x^{1/5}}+\frac{12y^{1/5}}{5}\times y'=0$$
Now we set all elements with $y'$ to one side and the rest to the other side: $$\frac{12y^{1/5}}{5}\times y'=-\frac{4}{5x^{1/5}}$$
Finally, we calculate $y'$: $$y'=-\frac{\frac{4}{5x^{1/5}}}{\frac{12y^{1/5}}{5}}=-\frac{20}{60x^{1/5}y^{1/5}}$$ $$y'=-\frac{1}{3(xy)^{1/5}}$$
