Step 1

Consider the given function \(\displaystyle{h}{\left({x},{y},{z}\right)}={\cos{{\left({x}+{y}+{z}\right)}}}\)

Step 2

Now find the partial derivative with respect to x

\(\displaystyle{\frac{{\partial}}{{\partial{x}}}}{\left({h}{\left({x},{y},{z}\right)}\right)}={\frac{{\partial}}{{\partial{x}}}}{\cos{{\left({x}+{y}+{z}\right)}}}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\frac{{\partial}}{{\partial{x}}}}{\left({x}+{y}+{z}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\left({1}+{0}+{0}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\)

Step 3

Now find the partial derivative with respect to y

\(\displaystyle{\frac{{\partial}}{{\partial{y}}}}{\left({h}{\left({x},{y},{z}\right)}\right)}={\frac{{\partial}}{{\partial{x}}}}{\cos{{\left({x}+{y}+{z}\right)}}}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\frac{{\partial}}{{\partial{y}}}}{\left({x}+{y}+{z}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\left({0}+{1}+{0}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\)

Step 4

Now find the partial derivative with respect to z

\(\displaystyle{\frac{{\partial}}{{\partial{z}}}}{\left({h}{\left({x},{y},{z}\right)}\right)}={\frac{{\partial}}{{\partial{x}}}}{\cos{{\left({x}+{y}+{z}\right)}}}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\frac{{\partial}}{{\partial{z}}}}{\left({x}+{y}+{z}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\left({0}+{0}+{1}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\)

Consider the given function \(\displaystyle{h}{\left({x},{y},{z}\right)}={\cos{{\left({x}+{y}+{z}\right)}}}\)

Step 2

Now find the partial derivative with respect to x

\(\displaystyle{\frac{{\partial}}{{\partial{x}}}}{\left({h}{\left({x},{y},{z}\right)}\right)}={\frac{{\partial}}{{\partial{x}}}}{\cos{{\left({x}+{y}+{z}\right)}}}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\frac{{\partial}}{{\partial{x}}}}{\left({x}+{y}+{z}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\left({1}+{0}+{0}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\)

Step 3

Now find the partial derivative with respect to y

\(\displaystyle{\frac{{\partial}}{{\partial{y}}}}{\left({h}{\left({x},{y},{z}\right)}\right)}={\frac{{\partial}}{{\partial{x}}}}{\cos{{\left({x}+{y}+{z}\right)}}}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\frac{{\partial}}{{\partial{y}}}}{\left({x}+{y}+{z}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\left({0}+{1}+{0}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\)

Step 4

Now find the partial derivative with respect to z

\(\displaystyle{\frac{{\partial}}{{\partial{z}}}}{\left({h}{\left({x},{y},{z}\right)}\right)}={\frac{{\partial}}{{\partial{x}}}}{\cos{{\left({x}+{y}+{z}\right)}}}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\frac{{\partial}}{{\partial{z}}}}{\left({x}+{y}+{z}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\left({0}+{0}+{1}\right)}\)

\(\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\)