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
ANSWERED
asked 2021-05-16
Find the partial derivatives:
\(\displaystyle{f{{\left({x},{y}\right)}}}={7}{x}^{{{7}\cdot{4}}}{y}^{{{3}}}+{z}^{{{3}}}\)
\(\displaystyle{\frac{{\partial{f}}}{{\partial{x}}}}=\)?
\(\displaystyle{\frac{{\partial{f}}}{{\partial{y}}}}=\)?
\(\displaystyle{\frac{{\partial{f}}}{{\partial{z}}}}=\)?

Answers (1)

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