1. ## Implicit Diff Problem. Whao made this stuff up

I have tried so many ways and I am sure I am over analyzing these problems but I dont know if I am suppose to use the Chain Rule Product Rule, I am becoming more confused and just need a step by step visual to get me jump started. If you have time could you please explain why each thing takes place. If not I understand, anyway here is the question.

Find DxY by Implicit Diff

xy + sin(xy) = 1

2. ## Re: Implicit Diff Problem. Whao made this stuff up

Originally Posted by psilver1
I have tried so many ways and I am sure I am over analyzing these problems but I dont know if I am suppose to use the Chain Rule Product Rule, I am becoming more confused and just need a step by step visual to get me jump started. If you have time could you please explain why each thing takes place. If not I understand, anyway here is the question.

Find DxY by Implicit Diff

xy + sin(xy) = 1
$\frac{d}{dx}(xy)+\frac{d}{dx}\sin(xy)=\frac{d}{dx} 1$

$y \frac{d}{dx}x+x\frac{d}{dx}y+\cos(xy)\frac{d}{dx}( xy)=0$

$y \frac{d}{dx}x+x\frac{d}{dx}y+\cos(xy)(y\frac{d}{dx }x+x\frac{d}{dx}y)=0$

$y+x\frac{dy}{dx}+\cos(xy)(y+x\frac{dy}{dx})=0$

Now just solve for the derivative.

3. ## Re: Implicit Diff Problem. Whao made this stuff up

For the first term, we may use the product rule, for the second term the chain rule, then the product rule, and the right side is of course zero:

$x\cdot\frac{dy}{dx}+y+\cos(xy)\left(x\cdot\frac{dy }{dx}+y \right)=0$

$\left(x\cdot\frac{dy}{dx}+y \right)(1+\cos(xy))=0$

What does this imply?

4. ## Re: Implicit Diff Problem. Whao made this stuff up

So (xy' + y)(xy'cos(xy) + ycos(xy))=0
sorta stuck does the xy' factor out. (xy' + y)(cos(xy) + ycos(xy))=0

thanks for the help so far. the above is where I am screwing this thing up. Mark I dont see where the +cos(xy)[x * dy/dx + y] turns into 1 + cos(xy). Btw I am sure you are right, i just dont understand where it comes from is what I am trying to say

5. ## Re: Implicit Diff Problem. Whao made this stuff up

Originally Posted by psilver1
So (xy' + y)(xy'cos(xy) + ycos(xy))=0
sorta stuck does the xy' factor out. (xy' + y)(cos(xy) + ycos(xy))=0

thanks for the help so far. the above is where I am screwing this thing up. Mark I dont see where the +cos(xy)[x * dy/dx + y] turns into 1 + cos(xy). Btw I am sure you are right, i just dont understand where it comes from is what I am trying to say
Look closely, Mark simply factored the whole thing. It's correct.

6. ## Re: Implicit Diff Problem. Whao made this stuff up

Ahh I see it. So is 1+cos(xy) and identity. Sorry for the questions just what to clear my mind.

7. ## Re: Implicit Diff Problem. Whao made this stuff up

xy+sin(xy)-1 as surface:

8. ## Re: Implicit Diff Problem. Whao made this stuff up

It is simply a factor we may ignore, as it doesn't involve $\frac{dy}{dx}$, and we know it cannot be zero, since this would imply:

$xy=(2k+1)\pi$

But, if this is true, then the original equation becomes:

$xy=1$