Question

If z= sinxy show that 1/y ∂z/∂x= 1/x ∂x/∂y​

Answer 1

$\text{Given that,}\\\\z = \sin(xy)\\\\\text{Now,}\\\\\textbf{L.H.S}\\\\=\dfrac 1y \cdot\dfrac{\partial z }{\partial x}\\\\\\=\dfrac{1}{y} \cdot \dfrac{\partial }{\partial x}(\sin(xy))\\\\\\=\dfrac 1y \cdot \cos(xy) \cdot y\dfrac{\partial }{\partial x}(x)\\\\\\=\cos(xy)$

$\textbf{R.H.S}\\\\=\dfrac 1x \cdot \dfrac{\partial z }{\partial y}\\\\\\=\dfrac 1x \cdot \dfrac{\partial }{\partial y}(\sin (xy))\\\\\\=\dfrac 1x \cdot x \cdot \cos(xy) \dfrac{\partial }{\partial y}(y)\\\\\\=\cos(xy)$

$\textbf{L.H.S} = \textbf{R.H.S}\\\\\text{Showed.}$