# Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables

Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables.
$$\displaystyle{w}={f{{\left({x},{y},{z}\right)}}}={\sin{{\left({x}+{y}-{z}\right)}}}$$

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Step 1
Consider the following function:
$$\displaystyle{w}={\sin{{\left({x}+{y}−{z}\right)}}}$$
Differentiate w with respect to x taking y and z as constant:
$$\displaystyle{\frac{{\partial{w}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\sin{{\left({x}+{y}-{z}\right)}}}$$
$$\displaystyle={\cos{{\left({x}+{y}-{z}\right)}}}$$
Differentiate w with respect to y taking x and z as constant:
$$\displaystyle{\frac{{\partial{w}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\sin{{\left({x}+{y}-{z}\right)}}}$$
$$\displaystyle={\cos{{\left({x}+{y}-{z}\right)}}}$$
Differentiate w with respect to z taking x and y as constant:
$$\displaystyle{\frac{{\partial{w}}}{{\partial{z}}}}={\frac{{\partial}}{{\partial{z}}}}{\sin{{\left({x}+{y}-{z}\right)}}}$$
$$\displaystyle=-{\cos{{\left({x}+{y}-{z}\right)}}}$$
Step 2
Consider the following formula:
$$\displaystyle{d}{w}={\frac{{\partial{w}}}{{\partial{x}}}}{\left.{d}{x}\right.}+{\frac{{\partial{w}}}{{\partial{y}}}}{\left.{d}{y}\right.}+{\frac{{\partial{w}}}{{\partial{z}}}}{\left.{d}{z}\right.}$$
$$\displaystyle={\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{x}\right.}+{\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{y}\right.}-{\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{z}\right.}$$