# derivative

• December 4th 2007, 06:12 PM
doctorgk
derivative
what is the second derivative of x^2 + y^2 = s^2

thanks
• December 4th 2007, 06:22 PM
topsquark
Quote:

Originally Posted by doctorgk
what is the second derivative of x^2 + y^2 = s^2

thanks

Ummm... You are looking for $\frac{d^2y}{dx^2}$? And I presume s is a constant?

$2x + 2y \frac{dy}{dx} = 0$

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

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

Insert the equation for $\frac{dy}{dx}$:
$\frac{d^2y}{dx^2} = -\frac{y - x \cdot -\frac{x}{y}}{y^2}$

$\frac{d^2y}{dx^2} = -\frac{y^2 + x^2}{y^3}$

Inserting the original equation:
$\frac{d^2y}{dx^2} = -\frac{s^2}{y^3}$

I'd probably leave it like that.

-Dan