# implict differential

• Feb 9th 2009, 11:21 PM
Arez
implict differential
please, someone help, how would i do $x^2 + 2(x^2)y + 3xy = 0$ with implicit differential..

also, how would i find $f'(-t cos(PI/t) (PI)$ (t^-2) $(sin(PI/2)))$ there are more than two terms

thanks
• Feb 9th 2009, 11:42 PM
earboth
Quote:

Originally Posted by Arez
please, someone help, how would i do $x^2 + 2(x^2)y + 3xy = 0$ with implicit differential..

...

If you have a function f(x,y) = 0 then you can differentiate it using the formula:

$f(x,y)=0~\implies~\dfrac{dy}{dx}=y'=-\dfrac{\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}}$

$y'=-\dfrac{2x+4xy+3y}{2x^2+3x}$

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If you don't want to use this formula differentiate the given equation wrt x:

$x^2 + 2(x^2)y + 3xy = 0~\implies~2x+(4x\cdot y + 2x^2 \cdot y') + (3y+3x\cdot y')=0$

Expand the brackets and collect all terms containing y' at the LHS:

$2x^2y'+3xy'=-2x-4xy-3y$

Factor out y' and divide both sides by the bracket. You'll get the result which you've got by the formula too.