More Implicit Differentiation

**Truthbetold** · November 29th 2007, 10:05 PM

$x^2+y^2=1$
Find dy/dx and the second derivative.

The answer says both derivatives equal $\frac{-x}{y}$ .
The first one does, but the second derivative I got as $\frac{y}{x}$ .

**Jhevon** · November 29th 2007, 10:12 PM

Originally Posted by Truthbetold

$x^2+y^2=1$
Find dy/dx and the second derivative.

The answer says both derivatives equal $\frac{-x}{y}$ .
The first one does, but the second derivative I got as $\frac{y}{x}$ .

and how did you get that answer?

i got neither. my answer for the second derivative was more complicated. i differentiated -x/y with the quotient rule. but you can use the product rule also

**kalagota** · November 30th 2007, 04:40 AM

aww, i got $\frac{-1}{y^3}$ ..

**colby2152** · November 30th 2007, 05:09 AM

$x^2 + y^2 = 1$
$2x\frac{dy}{dx} + 2y = 0$
$\frac{dy}{dx} = \frac{-y}{x}$

$\frac{d^2y}{dx^2} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{dy}{dx} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{-y}{x} = \frac{-1}{x} + \frac{-y^2}{x^3} = \frac{-(x^2+y^2)}{x^3}$

I got something different too. Hmm, did I miss something?

**kalagota** · November 30th 2007, 05:41 PM

Originally Posted by colby2152

$x^2 + y^2 = 1$
$2x\frac{dy}{dx} + 2y = 0$
$\frac{dy}{dx} = \frac{-y}{x}$

$\frac{d^2y}{dx^2} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{dy}{dx} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{-y}{x} = \frac{-1}{x} + \frac{-y^2}{x^3} = \frac{-(x^2+y^2)}{x^3}$

I got something different too. Hmm, did I miss something?

it's because, $\frac{dy}{dx} = \frac{-x}{y}$ and not $\frac{dy}{dx} = \frac{-y}{x}$ .. Ü

**galactus** · November 30th 2007, 05:54 PM

Originally Posted by colby2152

$x^2 + y^2 = 1$
$2x\frac{dy}{dx} + 2y = 0$
$\frac{dy}{dx} = \frac{-y}{x}$

$\frac{d^2y}{dx^2} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{dy}{dx} = \frac{-1}{x} + \frac{-y}{-x^2}\frac{-y}{x} = \frac{-1}{x} + \frac{-y^2}{x^3} = \frac{-(x^2+y^2)}{x^3}$

I got something different too. Hmm, did I miss something?

You're OK, except it should be $y^{3}$ in the denominator.

**colby2152** · November 30th 2007, 06:38 PM

Originally Posted by galactus

You're OK, except it should be $y^{3}$ in the denominator.

I see what I did, I flipped the derivative in the second line. It's easy to miss these things when typing them.

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