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:

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

\(\displaystyle={\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{x}\right.}+{\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{y}\right.}-{\cos{{\left({x}+{y}-{z}\right)}}}{\left.{d}{z}\right.}\)