Implicit differentiation from Larson 13.5

Question

Implicit differentiation: $$\frac{x}{x^{2}+y^{2}}-y^{2}=5$$ I've tried several ways including Wolfram, and the answer is not getting accepted. This is how I did it so far:

Answer 1

I do not know how implicit differentiation has been teached to you; so my naswer may be not the appropriate one.

Consider a function an implicit function $F(x,y)=0$ and consider each partial derivative $F'_x(x,y)$ and $F'_y(x,y)$. The implicit function theorem leads to $$\frac{dy}{dx}=-\frac{F'_x(x,y) } {F'_y(x,y) }$$ This is derived from the expression of the total derivative $$dF(x,y)=F'_x(x,y) \,dx+F'_y(x,y) \,dy=0$$

Let us apply to

$$F(x,y)=\frac{x}{x^{2}+y^{2}}-y^{2}-5=0$$ So, using the standard derivation steps, $$F'_x(x,y)=\frac{y^2-x^2}{\left(x^2+y^2\right)^2}$$ $$F'_y(x,y)=-\frac{2 x y}{\left(x^2+y^2\right)^2}-2 y$$ which then make, after some minor simplifications, $$\frac{dy}{dx}=-\frac{F'_x(x,y) } {F'_y(x,y) }=\frac{y^2-x^2}{2 y \left((x^2+y^2)^2+x)\right)}$$ which is exactly what your image shows.