Answer
$$dz = \frac{3}{{y + 3z}}dx + \frac{{12z - 3x}}{{{{\left( {y + 3z} \right)}^2}}}dy - \frac{{9x + 12y}}{{{{\left( {y + 3z} \right)}^2}}}dz$$
Work Step by Step
$$\eqalign{
& w = \frac{{3x + 4y}}{{y + 3z}} \cr
& {\text{The total differential of the dependent variable }}w{\text{ is}} \cr
& dw = \frac{{\partial w}}{{\partial x}}dx + \frac{{\partial w}}{{\partial y}}dy + \frac{{\partial w}}{{\partial z}}dz \cr
& {\text{Calculating }}\frac{{\partial w}}{{\partial x}}{\text{, }}\frac{{\partial w}}{{\partial y}}{\text{ and }}\frac{{\partial w}}{{\partial z}} \cr
& \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\frac{{3x + 4y}}{{y + 3z}}} \right] \cr
& \frac{{\partial w}}{{\partial x}} = \frac{\partial }{{\partial x}}\left[ {\frac{{3x}}{{y + 3z}} + \frac{{4y}}{{y + 3z}}} \right] \cr
& \frac{{\partial w}}{{\partial x}} = \frac{3}{{y + 3z}} + 0 \cr
& \frac{{\partial w}}{{\partial x}} = \frac{3}{{y + 3z}} \cr
& \cr
& \frac{{\partial w}}{{\partial y}} = \frac{\partial }{{\partial y}}\left[ {\frac{{3x + 4y}}{{y + 3z}}} \right] \cr
& {\text{By quotient rule}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{{\left( {y + 3z} \right)\left( 4 \right) - \left( {3x + 4y} \right)\left( 1 \right)}}{{{{\left( {y + 3z} \right)}^2}}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{{4y + 12z - 3x - 4y}}{{{{\left( {y + 3z} \right)}^2}}} \cr
& \frac{{\partial w}}{{\partial y}} = \frac{{12z - 3x}}{{{{\left( {y + 3z} \right)}^2}}} \cr
& \cr
& and \cr
& \cr
& \frac{{\partial w}}{{\partial z}} = \frac{\partial }{{\partial z}}\left[ {\frac{{3x + 4y}}{{y + 3z}}} \right] \cr
& \frac{{\partial w}}{{\partial z}} = - \frac{{\left( {3x + 4y} \right)\left( 3 \right)}}{{{{\left( {y + 3z} \right)}^2}}} \cr
& \frac{{\partial w}}{{\partial z}} = - \frac{{9x + 12y}}{{{{\left( {y + 3z} \right)}^2}}} \cr
& \cr
& {\text{Therefore,}} \cr
& dw = \frac{{\partial w}}{{\partial x}}dx + \frac{{\partial w}}{{\partial y}}dy + \frac{{\partial w}}{{\partial z}}dz \cr
& dz = \frac{3}{{y + 3z}}dx + \frac{{12z - 3x}}{{{{\left( {y + 3z} \right)}^2}}}dy - \frac{{9x + 12y}}{{{{\left( {y + 3z} \right)}^2}}}dz \cr} $$
