How do you find the second derivative of x 2 y 2 = 1

Question

How do you find the second derivative of             x      2              y      2        =    1

slapadabassyc · Accepted Answer

First we need to find the first derivative of the function.
We will do this using implicit differentiation. With this particular function we will use the product rule.

2 x y^{2} + x^{2} (2 y) (\frac{d y}{d x}) = 0

Now subtract

2 x y^{2}

from both sides

x^{2} (2 y) \frac{d y}{d x} = - 2 x y^{2}

Divide both sides by

x^{2} (2 y)

\frac{d y}{d x} = \frac{- 2 x y^{2}}{x^{2} (2 y)}

Which simplifies to

\frac{d y}{d x} = - \frac{y}{x}

Now for the second derivative we will use the quotient rule

\frac{d^{2} y}{{d x}^{2}} = \frac{(x (- 1) \frac{d y}{d x}) - (1 (- y))}{x^{2}}

\frac{d^{2} y}{{d x}^{2}} = \frac{- x \frac{d y}{d x} + y}{x^{2}} = \frac{y - x \frac{d y}{d x}}{x^{2}}

plug

\frac{d y}{d x}

into the right hand side

\frac{d^{2} y}{{d x}^{2}} = \frac{y - x (\frac{- y}{x})}{x^{2}} = \frac{y + y}{x^{2}} = \frac{2 y}{x^{2}}

\frac{d^{2} y}{{d x}^{2}} = \frac{2 y}{x^{2}}

How do you find the second derivative of x^2y^2=1

Answered question

Answer & Explanation

New Questions in Calculus 2