Answer
$dT=\dfrac{-v(vw)}{(1+uvw)^2}du+\dfrac{1+uvw-v(uw)}{(1+uvw)^2} dv+\dfrac{-v(uv)}{(1+uvw)^2} dw=\dfrac{1}{(1+uvw)^2}(-v^2w du+dv-uv^2dw)$
Work Step by Step
Here, we have
$T_u=\dfrac{(0v) (vw)}{(1+uvw)^2} ; \\T_v=\dfrac{1+uvw-v(uw)}{(1+uvw)^2} ; \\T_w=\dfrac{(-v) (uv)}{(1+uvw)^2}$
$dT=\dfrac{\partial T}{\partial u} du +\dfrac{\partial T}{\partial v} dv+ \dfrac{\partial T}{\partial w} dw \\=\dfrac{-v(vw)}{(1+uvw)^2}du+\dfrac{1+uvw-v(uw)}{(1+uvw)^2} dv+\dfrac{-v(uv)}{(1+uvw)^2} dw \\=\dfrac{1}{(1+uvw)^2}(-v^2w du+dv-uv^2dw)$
