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}\\\\=\frac 1y \cdot(\partial z )/(\partial x)\\\\\\=(1)/(y) \cdot (\partial )/(\partial x)(\sin(xy))\\\\\\=\frac 1y \cdot \cos(xy) \cdot y(\partial )/(\partial x)(x)\\\\\\=\cos(xy)`

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

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