Step 1

The function is \(\displaystyle{z}={x}{y}+{x}{\left({y}^{{2}}+{1}\right)}\)

find the partial derivatives

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={y}+{\left({y}^{{2}}+{1}\right)}\)

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={x}+{x}{\left({2}{y}\right)}\)

The partial derivatives are:

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={y}+{y}^{{2}}+{1}\)

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={x}+{2}{x}{y}\)

Step 2

total differential is given by \(\displaystyle{\left.{d}{z}\right.}=\frac{{\partial{z}}}{{\partial{x}}}{\left.{d}{x}\right.}+\frac{{\partial{z}}}{{\partial{y}}}{\left.{d}{y}\right.}\)

Substitute the values

\(\displaystyle{\left.{d}{z}\right.}={\left({y}+{y}^{{2}}+{1}\right)}{\left.{d}{x}\right.}+{\left({x}+{2}{x}{y}\right)}{\left.{d}{y}\right.}\)

The total differential is:

\(\displaystyle{\left.{d}{z}\right.}={\left({y}+{y}^{{2}}+{1}\right)}{\left.{d}{x}\right.}+{\left({x}+{2}{x}{y}\right)}{\left.{d}{y}\right.}\)

Step 3

The total derivative with respect to x is given by

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}=\frac{{\partial{z}}}{{\partial{x}}}+\frac{{\partial{z}}}{{\partial{y}}}\frac{{\partial{y}}}{{\partial{x}}}\)

Substitute the values

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={\left({y}+{y}^{{2}}+{1}\right)}+{\left({x}+{2}{x}{y}\right)}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}\)

The total derivative with respect to x is:

\(\displaystyle\frac{{{\left.{d}{z}\right.}}}{{{\left.{d}{x}\right.}}}={\left({y}+{y}^{{2}}+{1}\right)}+{\left({x}+{2}{x}{y}\right)}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}\)

The function is \(\displaystyle{z}={x}{y}+{x}{\left({y}^{{2}}+{1}\right)}\)

find the partial derivatives

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={y}+{\left({y}^{{2}}+{1}\right)}\)

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={x}+{x}{\left({2}{y}\right)}\)

The partial derivatives are:

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={y}+{y}^{{2}}+{1}\)

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={x}+{2}{x}{y}\)

Step 2

total differential is given by \(\displaystyle{\left.{d}{z}\right.}=\frac{{\partial{z}}}{{\partial{x}}}{\left.{d}{x}\right.}+\frac{{\partial{z}}}{{\partial{y}}}{\left.{d}{y}\right.}\)

Substitute the values

\(\displaystyle{\left.{d}{z}\right.}={\left({y}+{y}^{{2}}+{1}\right)}{\left.{d}{x}\right.}+{\left({x}+{2}{x}{y}\right)}{\left.{d}{y}\right.}\)

The total differential is:

\(\displaystyle{\left.{d}{z}\right.}={\left({y}+{y}^{{2}}+{1}\right)}{\left.{d}{x}\right.}+{\left({x}+{2}{x}{y}\right)}{\left.{d}{y}\right.}\)

Step 3

The total derivative with respect to x is given by

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}=\frac{{\partial{z}}}{{\partial{x}}}+\frac{{\partial{z}}}{{\partial{y}}}\frac{{\partial{y}}}{{\partial{x}}}\)

Substitute the values

\(\displaystyle\frac{{\partial{z}}}{{\partial{x}}}={\left({y}+{y}^{{2}}+{1}\right)}+{\left({x}+{2}{x}{y}\right)}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}\)

The total derivative with respect to x is:

\(\displaystyle\frac{{{\left.{d}{z}\right.}}}{{{\left.{d}{x}\right.}}}={\left({y}+{y}^{{2}}+{1}\right)}+{\left({x}+{2}{x}{y}\right)}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}\)