# Why does this implicit differentiation formula fail?

Suppose we have that

$$\frac{dy}{dx} = -\frac{y}{x}.$$

Taking the derivative implicitly with respect to $x$, we can easily obtain

$$\frac{d^{2}y}{dx^{2}} = \frac{-\frac{dy}{dx}x + y}{x^{2}} = \frac{2y}{x^{2}}.$$

I figured that if I wanted to find $\frac{d^{2}y}{dx^{2}}$, I should be able to simply set $F = -y/x$ and get

$$\frac{d^{2}y}{dx^{2}} = - \frac{F_{x}}{F_{y}} = \frac{\frac{y}{x^{2}}}{\frac{1}{x}} = \frac{y}{x}$$

which clearly does not work. I tried taking the total derivative of

$$\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y}\frac{dy}{dx} = 0$$

and it seems pretty clear that what I did was unlikely to work, but I don't understand why my intuition in this case would fail since $\frac{dy}{dx}$ is just a function that we can implicitly differentiate.

if you wanted to go the long way..by setting $F = -y/x$ then you would have

$$y'' = \frac{\partial F}{\partial x} + \frac{\partial F}{\partial y}y'$$ or $$y'' = \frac{y}{x^2} +\left(-\frac{1}{x}\right)\left(-\frac{y}{x}\right) = \frac{2y}{x^2}$$ now the problem is that you have the original result wrong it should not have a minus sign

• opps, too many negatives signs always confuses me – JessicaK Nov 26 '14 at 11:08
• No worries! I do the exact same, but I always have to start from scratch even if I have already done 4 pages of calculations .. Damn OCD! – Chinny84 Nov 26 '14 at 11:10
• After looking at these answers I completely understand what to do, but I can't help but feel disappointed more than anything that I cannot take $-F_{x}/F_{y}$ in this case. Thank you. – JessicaK Nov 26 '14 at 11:27

If you take $$\frac{dy}{dx} = -\frac{y}{x} = F(y,x)$$

This equation $$\frac{d^{2}y}{dx^{2}} = - \frac{F_{x}}{F_{y}}$$ is false. You should write

$$\frac{d^{2}y}{dx^{2}} = \frac {d}{dx}F(y(x),x) = \frac{\partial F}{\partial y} \frac{dy}{dx} +\frac{\partial F}{\partial x} = \left(-\frac 1x \right)\left(-\frac yx\right)+\frac{y}{x^2} = \frac{2y}{x^2}.$$