1. ## Derivative?

If $y = \frac{\sin x}{1 + \frac{\cos x}{1 + \frac{\sin x}{1 + \frac{\cos x}{1 + \sin x\mbox{...}\infty}}}}$, then prove that $\frac{dy}{dx} = \frac{(1 + y)\cos x + y\sin x}{1 + 2y + \cos x - \sin x}$

2. $y=\frac{sin x}{1+\frac{cos x}{1+y}}$
$y=\frac{sin x(1+y)}{1+y+cos x}$
$y+y^2+ycos x=sin x+ysin x$
$\frac{dy}{dx}+2y\frac{dy}{dx}+y*-sin x+cos x*\frac{dy}{dx}=cos x+ycos x+sin x\frac{dy}{dx}$
$\frac{dy}{dx}(1+2y+cos x)-ysin x=cos x(1+y)+sin x\frac{dy}{dx}$
$\frac{dy}{dx}(1+2y+cos x-sin x)=(1+y)cos x+y sin x$
$\frac{dy}{dx}(1+2y+cos x-sin x)=(1+y)cos x+y sin x$
$\frac{dy}{dx}=\frac{(1+y)cos x+y sin x}{1+2y+cos x-sin x}$

3. I'd like to see the calculus textbook from which you get these unusual (but interesting) problems.

4. Originally Posted by Random Variable
I'd like to see the calculus textbook from which you get these unusual (but interesting) problems.

5. Originally Posted by fardeen_gen
$y = \frac{\sin x}{1 + \frac{\cos x}{1 + \frac{\sin x}{1 + \frac{\cos x}{1 + \sin x\mbox{...}\infty}}}}$
To write continued fractions one can use \cfrac instead of \frac, it makes the fraction more readable.

$y = \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \ldots}}}}}}}}$

6. Originally Posted by flyingsquirrel
To write continued fractions one can use \cfrac instead of \frac, it makes the fraction more readable.

$y = \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \cfrac{\sin x}{1 + \cfrac{\cos x}{1 + \ldots}}}}}}}}$
I wanted to create a new thread asking exactly this - how do write continued fractions. Thanks a lot