Thread: Implicit differenriarion

Hello All,

I'm working on differentiations right now and I'm having a little trouble trying to fully understand. I did this one, but I'm not sure if it's right.

$\displaystyle
x^2y=x+y^2$

$\displaystyle \frac{d}{dx}[x^2y]=\frac{d}{dx}[x]+\frac{d}{dx}[y^2]$

$\displaystyle 2xy\frac{dy}{dx}[y]=x+\frac{dy}{dx}[2y]$

$\displaystyle
\frac{dy}{dx}(y-2y)=x-2xy$

$\displaystyle
\frac{dy}{dx}=\frac{x-2xy}{y-2y}$

Did I do it right?

Originally Posted by drkidd22

Hello All,

I'm working on differentiations right now and I'm having a little trouble trying to fully understand. I did this one, but I'm not sure if it's right.

$\displaystyle
x^2y=x+y^2$

$\displaystyle \frac{d}{dx}[x^2y]=\frac{d}{dx}[x]+\frac{d}{dx}[y^2]$ Good

$\displaystyle 2xy\frac{dy}{dx}[y]=x+\frac{dy}{dx}[2y]$ Product rule on left hand side... and, what is the derivative of x with respect to x?

$\displaystyle
\frac{dy}{dx}(y-2y)=x-2xy$

$\displaystyle
\frac{dy}{dx}=\frac{x-2xy}{y-2y}$

Did I do it right?

Sorry I can't get the red to work, will post more in a tic

Edit:

Just in case a picture helps...



... where



... is the product rule, and



the chain rule. (Straight continuous lines differentiate downwards with respect to x, the straight dashed line similarly but with respect to the dashed balloon expression which is the inner function for the chain rule.)


_________________________________
Don't integrate - balloontegrate!

http://www.ballooncalculus.org/forum/top.php

Draw balloons with Latex: http://www.ballooncalculus.org/asy/doc.html

The LHS is incorrect--use product rule:

d(x^2y)/dx = 2xy + x^2dy/dx

The Rhs is 1 +2ydy/dx


2xy + x^2dy/dx = 1 +2ydy/dx


now finish

In general for implicit differentiation

1. differentiate fns of x normally d(x^2)/dx = 2x

2. when differentiating fns of y diff with respect to y and multiply by dy/x

d(y^2)/dx = 2y dy/dx

$\displaystyle \frac{dy}{dx}=\frac{1-2xy}{x^2-2y}$

Yep

thanks, I liked the diagram you posted. Look my other thread coming up. I got another one.