While solving the sums from my textbook on implicit differentiation, I happened to encounter a sum :
$$ Y^3 + XY + X^2 = 0 $$
and I was suppose to find dY/dX and substitute the value 10 for X after differentiating. Weirdly enough, this equation does not let me get rid of the variable Y after solving and hence I couldn't find a valid answer.
Would request you to help me approach the problem. Also, I would like to do so without finding the roots of Y and solving, as someone pointed out earlier in class.
Sorry for the aweful quality.
Let $F=y^3+x y+x^2$. Then $$F'_x=y+2x$$ $$F'_y=3y^2+x$$ $$\frac{dy}{dx}=-\frac{F'_x}{F'_y}=-\frac{y+2x}{3y^2+x}$$ We know $x=10$ and we need the corresponding $y$ which is the solution of $y^3+10y+100=0$ which is cubic equation that you can solve using Cardano. This equation has a single real root given by $$y=\frac{\sqrt[3]{10 \left(\sqrt{2055}-45\right)}}{3^{2/3}}-\frac{10^{2/3}}{\sqrt[3]{3 \left(\sqrt{2055}-45\right)}}\simeq -3.93003$$ If you do not use Cardano, only a numerical method could be used. The simplest would be Newton, which, starting from a "reason able" guess $y_0$ will update it according to $$y_{n+1}=y_n-\frac{f(y_n)}{f'(y_n)}$$ with $f(y)=y^3+10y+100$. By inspection, you could see that there is a root close to $-4$; so, let us select $y_0=-4$. Soo, Newton iterates will be $-3.93103$,$-3.93003$.