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# Consider a function z = xy+x(y^2+1).Find first order partial derivatives, total differential, and total derivative with respect to x.

Question
Derivatives
asked 2021-02-09
Consider a function $$\displaystyle{z}={x}{y}+{x}{\left({y}^{{2}}+{1}\right)}$$.Find first order partial derivatives, total differential, and total derivative with respect to x.

## Answers (1)

2021-02-10
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:
$$\displaystyle\frac{{{\left.{d}{z}\right.}}}{{{\left.{d}{x}\right.}}}={\left({y}+{y}^{{2}}+{1}\right)}+{\left({x}+{2}{x}{y}\right)}\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{x}\right.}}}$$

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