# Math Help - [SOLVED] Implicit Differentiation

1. ## [SOLVED] Implicit Differentiation

$y^2=cos(xy)+ln(x+y)$

Find $\frac{dy}{dx}$ in terms of $x$ and $y$

Now, this is what I did, I'm not entirely sure if I'm doing it right.

$2y.\frac{dy}{dx} = -ysin(xy).(y+x\frac{dy}{dx})+\frac{1}{x} + \frac{1}{y}\frac{dy}{dx}$

$\frac{-1}{x}+ysin(xy)=(\frac{1}{y}-xsin(xy)-2y)\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{ysin(xy)-\frac{1}{x}}{\frac{1}{xy}-xsin(xy)-2y}$

So do i get or ?

2. ## My working is

2y*dy=-sin(xy)*(y+x*dy)+(1+dy)/(x+y)
where dy=dy/dx
solve dy you'll get the answer?

3. Originally Posted by Rapid_W
$y^2=cos(xy)+ln(x+y)$

Find $\frac{dy}{dx}$ in terms of $x$ and $y$

Now, this is what I did, I'm not entirely sure if I'm doing it right.

$2y.\frac{dy}{dx} = {\color{blue}-ysin(xy).(y+x\frac{dy}{dx})}+{\color{red}\frac{1}{ x} + \frac{1}{y}\frac{dy}{dx}}$

$\frac{-1}{x}+ysin(xy)=(\frac{1}{y}-xsin(xy)-2y)\frac{dy}{dx}$

$\frac{dy}{dx}=\frac{ysin(xy)-\frac{1}{x}}{\frac{1}{xy}-xsin(xy)-2y}$

So do i get or ?
can you find your mistake in the blue and the red term or
want I tell you about them

4. Not really sure about the blue, but is the red supposed to be

$\frac{1}{xy}\frac{dy}{dx}$?

5. $ln(x+y)...derive...\frac{1}{x+y}+\frac{dy/dx}{y+x}$

6. ahh ok, should the blue be this?

$-ysin(xy)-xsin(xy).\frac{dy}{dx}$

7. yeah that's right you are correct

8. ok so starting over doing it correctly i get

$2y\frac{dy}{dx} = -sin(xy).y - sin(xy).x\frac{dy}{dx} + \frac{1}{x+y} + \frac{1}{x+y}\frac{dy}{dx}$
$
(2y + xsin(xy) - \frac{1}{x+y})\frac{dy}{dx} = -ysin(xy) + \frac{1}{x+y}$

$\frac{dy}{dx} = \frac{-ysin(xy)+(x+y)^{-1}}{2y+xsin(xy)-(x+y^{-1}}$

which simplifys to

$\frac{dy}{dx} = \frac{-ysin(xy)}{2y+xsin(xy)} -1$

Is this correct?

9. Originally Posted by Rapid_W
ok so starting over doing it correctly i get

$2y\frac{dy}{dx} = -sin(xy).y - sin(xy).x\frac{dy}{dx} + \frac{1}{x+y} + \frac{1}{x+y}\frac{dy}{dx}$
$
(2y + xsin(xy) - \frac{1}{x+y})\frac{dy}{dx} = -ysin(xy) + \frac{1}{x+y}$

$\frac{dy}{dx} = \frac{-ysin(xy)+(x+y)^{-1}}{2y+xsin(xy)-(x+y^{-1}}$

which simplifys to

$\frac{dy}{dx} = \frac{-ysin(xy)}{2y+xsin(xy)} -1$

Is this correct?

$\frac{dy}{dx} = \frac{-ysin(xy)+(x+y)^{-1}}{2y+xsin(xy)-(x+y)^{-1}}$ this is correct

but how that became to this

$\frac{dy}{dx} = \frac{-ysin(xy)}{2y+xsin(xy)}-1$

I am not sure if it is wrong or not

$\frac{dy}{dx} = \frac{-ysin(xy)+(x+y)^{-1}}{2y+xsin(xy)-(x+y)^{-1}}$

$\frac{dy}{dx} =\left(\dfrac{ \dfrac{(x+y)(-ysin(xy))+1 }{(x+y)} }{\dfrac{ (x+y)(2y+xsin(xy)-1)}{(x+y)}}\right)$

$\frac{dy}{dx} =\left(\dfrac{ (x+y)(-ysin(xy))+1 }{(x+y)(2y+xsin(xy)-1)}\right)$

simplify it

10. ahh ok, that makes more sense, ok, my problem is now solved thanks everyone that helped