 Differentials with more than two variables Write the differential dw in terms of the differentials of the independent variables e1s2kat26 2021-09-07 Answered
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:
$$\displaystyle{d}{w}={\frac{{\partial{w}}}{{\partial{x}}}}{\left.{d}{x}\right.}+{\frac{{\partial{w}}}{{\partial{y}}}}{\left.{d}{y}\right.}+{\frac{{\partial{w}}}{{\partial{z}}}}{\left.{d}{z}\right.}$$
$$\displaystyle={\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{x}\right.}+{\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{y}\right.}-{\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{z}\right.}$$