Question

# Find the partial derivatives: f(x,y)=7x^{7*4}y^{3}+z^{3} \frac{\partial f}{\partial x}=? \frac{\partial f}{\partial y}=? \frac{\partial f}{\partial z}=?

Derivatives
Find the partial derivatives:
$$\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{{7}\cdot{4}}}{y}^{{{3}}}+{z}^{{{3}}}$$
$$\displaystyle{\frac{{\partial{f}}}{{\partial{x}}}}=$$?
$$\displaystyle{\frac{{\partial{f}}}{{\partial{y}}}}=$$?
$$\displaystyle{\frac{{\partial{f}}}{{\partial{z}}}}=$$?

2021-05-18
Step 1
The partial derivatives of function f are given below :
Step 2
$$\displaystyle{f{{\left({x},{y},{z}\right)}}}={7}\cdot{x}^{{{7}\cdot{4}}}\cdot{y}^{{{3}}}+{z}^{{{3}}}$$
$$\displaystyle{\frac{{\partial{f}}}{{\partial{x}}}}={\frac{{\partial}}{{\partial{x}}}}{\left[{7}\cdot{x}^{{{7}\cdot{4}}}\cdot{y}^{{{3}}}+{z}^{{{3}}}\right]}$$
$$\displaystyle={7}{\left({7}\cdot{4}\right)}{x}^{{{6}\cdot{4}}}{y}^{{{3}}}={51}\cdot{8}{x}^{{{6}\cdot{4}}}{y}^{{{3}}}$$
$$\displaystyle{\frac{{\partial{f}}}{{\partial{y}}}}={\frac{{\partial}}{{\partial{y}}}}{\left[{7}\cdot{x}^{{{7}\cdot{4}}}\cdot{y}^{{{3}}}+{z}^{{{3}}}\right]}={27}\cdot{x}^{{{7}\cdot{4}}}{y}^{{{2}}}$$
$$\displaystyle{\frac{{\partial{f}}}{{\partial{z}}}}={\frac{{\partial}}{{\partial{z}}}}{\left[{7}\cdot{x}^{{{7}\cdot{4}}}\cdot{y}^{{{3}}}+{z}^{{{3}}}\right]}={3}{z}^{{{2}}}$$