Question

Solve using implicit differentiation.
In terms of x and y.

Answer 1

\[Find \frac{ d^{2}y }{ dx^{2} }\]\[x^{2}y^{2}-2x=3\]

Answer 2

Having trouble with all of the differentiation or just finding the second?

Answer 3

Finding the second.

Answer 4

Okay, so i'll jump to the first without any work:

\[ \frac{dy}{dx} = \frac{1-xy^2}{yx^2} \]

Apply the differential operator to each side, (Right Hand Side will be the quotient rule with a bunch of products and chains)

\[\frac{d}{dx} (\frac{dy}{dx}) = \frac{d}{dx}(\frac{1-xy^2}{yx^2}) \]

\[ (\frac{d^2y}{dx^2}) = (\frac{(-y^2-2xy\frac{dy}{dx})yx^2-(2xy+x^2\frac{dy}{dx})(1-xy^2)}{(yx^2)^2}) \]

But recall, we know dy/dx, substitute it back into your equation

\[ (\frac{d^2y}{dx^2}) = \frac{(-y^2-2xy(\frac{1-xy^2}{yx^2}))yx^2-(2xy+x^2(\frac{1-xy^2}{yx^2}))(1-xy^2)}{(yx^2)^2} \]

You can simplify, but at this point, I would just walk away.

Answer 5

Thank you for your help.