# Find the first partial derivatives of the following functions. h(x,y,z)=\cos(x+y+z)

Find the first partial derivatives of the following functions.
$$\displaystyle{h}{\left({x},{y},{z}\right)}={\cos{{\left({x}+{y}+{z}\right)}}}$$

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StrycharzT
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
$$\displaystyle{\frac{{\partial}}{{\partial{z}}}}{\left({h}{\left({x},{y},{z}\right)}\right)}={\frac{{\partial}}{{\partial{x}}}}{\cos{{\left({x}+{y}+{z}\right)}}}$$
$$\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\frac{{\partial}}{{\partial{z}}}}{\left({x}+{y}+{z}\right)}$$
$$\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}\cdot{\left({0}+{0}+{1}\right)}$$
$$\displaystyle=-{\sin{{\left({x}+{y}+{z}\right)}}}$$
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