Find the first partial derivatives of the following functions. f(x,y)=xe^y

Find the first partial derivatives of the following functions. f(x,y)=xe^y

Question
Derivatives
asked 2021-03-09
Find the first partial derivatives of the following functions. \(\displaystyle{f{{\left({x},{y}\right)}}}={x}{e}^{{y}}\)

Answers (1)

2021-03-10
Step 1
To find the first-order partial derivatives of the function,
\(\displaystyle{f{{\left({x},{y}\right)}}}={x}{e}^{{y}}\)
Step 2
The partial derivatives with respect to x is given by,
\(\displaystyle{{f}_{{x}}{\left({x},{y}\right)}}=\lim_{{{h}\rightarrow{0}}}\frac{{{f{{\left({x}+{h},{y}\right)}}}-{f{{\left({x},{y}\right)}}}}}{{h}}=\lim_{{{h}\rightarrow{0}}}\frac{{{\left({x}+{h}\right)}{e}^{{y}}-{x}{e}^{{y}}}}{{h}}=\lim_{{{h}\rightarrow{0}}}\frac{{{x}{e}^{{y}}+{h}{e}^{{y}}-{x}{e}^{{y}}}}{{h}}=\lim_{{{h}\rightarrow{0}}}\frac{{{h}{e}^{{y}}}}{{h}}\)
\(\displaystyle=\lim_{{{h}\rightarrow{0}}}{e}^{{y}}={e}^{{y}}\)
The partial derivatives with respect to y is given by,
\(\displaystyle{{f}_{{y}}{\left({x},{y}\right)}}=\lim_{{{k}\rightarrow{0}}}\frac{{{f{{\left({x},{y}+{k}\right)}}}-{f{{\left({x},{y}\right)}}}}}{{h}}=\lim_{{{k}\rightarrow{0}}}\frac{{{x}{e}^{{{y}+{k}}}-{x}{e}^{{y}}}}{{k}}=\lim_{{{k}\rightarrow{0}}}\frac{{{x}{e}^{{y}}{\left({e}^{{k}}-{1}\right)}}}{{k}}={x}{e}^{{y}}\lim_{{{k}\rightarrow{0}}}\frac{{{e}^{{k}}-{1}}}{{k}}\)
\(\displaystyle={x}{e}^{{y}}\lim_{{{k}\rightarrow{0}}}{e}^{{k}}={x}{e}^{{y}}{\left({1}\right)}={x}{e}^{{y}}\)
0

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