Implicit Differentiation

May 2010
36
0
[FONT=&quot]Use implicit differentiation to find [/FONT][FONT=&quot]dy/dx for xy2 - yx2 = 3xy.

My working:[/FONT]
\(\displaystyle x.(2y.dy/dx)+y^2(1)-y(2x)+dy/dx(x^2)=3x dy/dx + 3y\)

\(\displaystyle ==> 2xy dy/dx + y^2 - 2xy + dy/dx (x^2) = 3x dy/dx + 3y\)


Now Im very confused. Do we need to divide both sides, in order to separate dy/dx? Any helpful tips/suggestions would be appreciated

Thanks!:)
[FONT=&quot]
[/FONT]
 
Last edited by a moderator:
Feb 2008
40
6
So \(\displaystyle xy^2 - x^2y = 3xy\)

Your working looks right although it's a little messy and I think youre dropping a sign somewhere so allow me:

\(\displaystyle y^2 + 2xy\frac{dy}{dx} - x^2\frac{dy}{dx} - 2xy = 3x\frac{dy}{dx} + 3y\)

If we throw \(\displaystyle \frac{dy}{dx}\)'s to one side and the rest to the other side:

\(\displaystyle y^2 - 2xy - 3y = 3x\frac{dy}{dx} - 2xy\frac{dy}{dx} + x^2\frac{dy}{dx} \)

Take out \(\displaystyle \frac{dy}{dx}\) as factor on RHS

\(\displaystyle y^2 - 2xy - 3y = \frac{dy}{dx} (3x - 2xy + x^2) \)

Divide both sides by \(\displaystyle (3x - 2xy + x^2)\) and you end up with:

\(\displaystyle \frac{y^2 - 2xy - 3y}{3x - 2xy + x^2} = \frac{dy}{dx}\)
 
May 2010
36
0
So \(\displaystyle xy^2 - x^2y = 3xy\)
Im sorry, I didn't type the question in proper notation. it's actually:
\(\displaystyle xy^2-yx^2=3xy\)

could you please help me with that? I'm still a little confused.

Thanks! :)
 
Feb 2008
40
6
Im sorry, I didn't type the question in proper notation. it's actually:
\(\displaystyle xy^2-yx^2=3xy\)

could you please help me with that? I'm still a little confused.

Thanks! :)
It shouldn't make a difference :)
 
May 2010
36
0
\(\displaystyle y^2 + 2xy\frac{dy}{dx} - x^2\frac{dy}{dx} - 2xy = 3x\frac{dy}{dx} + 3y\)
I actually got \(\displaystyle + x^2\frac{dy}{dx}\), using the product rule/chain rule of derivative.

is mine incorrect?
 
Feb 2008
40
6
That was the sign I mentioned you dropped earlier.

Look at it this way:

\(\displaystyle xy^2 - x^2y = 3xy \)


\(\displaystyle (y^2 + 2xy\frac{dy}{dx}) - (x^2\frac{dy}{dx} + 2xy) = 3x\frac{dy}{dx} + 3y\)

\(\displaystyle y^2 + 2xy\frac{dy}{dx} - x^2\frac{dy}{dx} - 2xy = 3x\frac{dy}{dx} + 3y\)


Does that make sense?
 
  • Like
Reactions: spoc21