I was asked to implicitly differentiate (x^2)y + y = 2, which I did:

use product rule: [fg]' = f'g + fg'
f=x^2
g=y
f'=2x
g'=y'

[(x^2)y + y]' = 2'
[(x^2)y]' + [y]' = 2'
2xy +(x^2)y' + y' = 0
2xy = -(x^2)y' - y'
2xy = -y'(x^2 + 1)
2xy
y' = - -------
x^2 +1
which matches the book's answer.

But then I thought why can't I solve for y and then differentiate explicitly?

I tried:
(x^2)y + y = 2 ==> y( (x^2) + 1) = 2 ===> y= 2/((x^2) + 1)

But my graphing program says
4x
y' = - -------------
(x^2 +1)^2

What's wrong?

It's the same, you just didn't sub in your y.

From implicit diff. we get \frac{dy}{dx}=\frac{-2xy}{x^{2}+1}

From doing it the 'regular' way, we get \frac{-4x}{(x^{2}+1)^{2}}

Now, when we solved the equation for y we got y=\frac{2}{x^{2}+1}

Sub that into your implicit derivative:

\frac{-2x(\frac{2}{x^{2}+1})}{x^{2}+1}=\frac{-4x}{(x^{2}+1)^{2}}

See?

both answers are the same even they look different

using explicit diff y' = -4x/(x^2+1)^2

using implicit diff y' = -2xy/(x^2+1) but the replacing y in the answer by 2/(x^2+1)
well get you the same above y=-4x/(x^2+1)^2






I was asked to implicitly differentiate (x^2)y + y = 2, which I did:

use product rule: [fg]' = f'g + fg'
f=x^2
g=y
f'=2x
g'=y'

[(x^2)y + y]' = 2'
[(x^2)y]' + [y]' = 2'
2xy +(x^2)y' + y' = 0
2xy = -(x^2)y' - y'
2xy = -y'(x^2 + 1)
2xy
y' = - -------
x^2 +1
which matches the book's answer.

But then I thought why can't I solve for y and then differentiate explicitly?

I tried:
(x^2)y + y = 2 ==> y( (x^2) + 1) = 2 ===> y= 2/((x^2) + 1)

But my graphing program says
4x
y' = - -------------
(x^2 +1)^2

What's wrong?