# Math Help - differentiate y = Dsin(kx/y) wrt x

1. ## differentiate y = Dsin(kx/y) wrt x

Given a function y = Dsin(kx/y) ,where D and k are constants and (kx/y) is in radians
1. what is dy/dx ?
2. how can you express x as a function of y , x=?
3. what is dx/dy ?

2. You will need to use the chain rule and implicit differentiation...

$y = D\sin{\frac{kx}{y}}$

Let $u = \frac{kx}{y}$ so that $y= D\sin{u}$.

$\frac{dy}{du} = D\cos{u} = D\cos{\left(\frac{kx}{y}\right)}$.

$\frac{du}{dx} = \frac{d}{dx}\left(\frac{kx}{y}\right)$

$= k\,\frac{d}{dx}\left(\frac{x}{y}\right)$

$= k\left(\frac{y\,\frac{d}{dx}(x) - x\,\frac{d}{dx}(y)}{y^2}\right)$

$= \frac{k\left(y - x\,\frac{dy}{dx}\right)}{y^2}$.

So $\frac{dy}{dx} = \frac{dy}{du}\,\frac{du}{dx}$

$\frac{dy}{dx} = \frac{Dk\left(y - x\,\frac{dy}{dx}\right)\cos{\left(\frac{kx}{y}\rig ht)}}{y^2}$

$y^2\,\frac{dy}{dx} = Dky\cos{\left(\frac{kx}{y}\right)} - Dkx\cos{\left(\frac{kx}{y}\right)}\frac{dy}{dx}$

$y^2\,\frac{dy}{dx} + Dkx\cos{\left(\frac{kx}{y}\right)}\frac{dy}{dx} = Dky\cos{\left(\frac{kx}{y}\right)}$

$\frac{dy}{dx}\left[y^2 + Dkx\cos{\left(\frac{kx}{y}\right)}\right] = Dky\cos{\left(\frac{kx}{y}\right)}$

$\frac{dy}{dx} = \frac{Dky\cos{\left(\frac{kx}{y}\right)}}{y^2 + Dkx\cos{\left(\frac{kx}{y}\right)}}$

3. ## Thanks

Thanks for the assistance, much appreciated.