Question

Find the four second partial derivatives of the following function.z=3ye^{7x}z_{xx}=ZKS?

Derivatives
ANSWERED
asked 2021-04-14

Find the four second partial derivatives of the following function.
\(\displaystyle{z}={3}{y}{e}^{{{7}{x}}}\)
\(z_{xx}=\)?

Expert Answers (1)

2021-04-16
Step 1
The four-second partial derivatives of function z=f(x,y),
\(\displaystyle{z}_{{\times}},{z}_{{{y}{y}}},{z}_{{{x}{y}}},{z}_{{{y}{x}}}\)
Step 2
Given: \(\displaystyle{z}={3}{y}{e}^{{{7}{x}}}\)(i)
Differentiate both sides with respect to x, we get
\(\displaystyle{z}_{{{x}}}={3}{y}{\left({7}\right)}{e}^{{{7}{x}}}={21}{y}{e}^{{{7}{x}}}\)(ii)
\(\displaystyle{z}_{{\times}}={21}{y}{\left({7}\right)}{e}^{{{7}{x}}}\)
i.e. \(\displaystyle{z}_{{\times}}={147}{y}{e}^{{{7}{x}}}\)
Differentiate (ii) with respect to y, we get
\(\displaystyle{z}_{{{y}}}={3}{e}^{{{7}{x}}}\)(iii)
\(\displaystyle\Rightarrow{z}_{{{y}{y}}}={0}\)
Differentiate (iii) with respect to x , we get
\(\displaystyle{z}_{{{y}{x}}}={3}{\left({7}\right)}{e}^{{{7}{x}}}={21}{e}^{{{7}{x}}}\)
i.e. \(\displaystyle{z}_{{{y}{x}}}={21}{e}^{{{7}{x}}}\)
Step 3
The four-second partial derivatives:
\(\displaystyle{z}_{{\times}}={147}{y}{e}^{{{7}{x}}}\)
\(\displaystyle{z}_{{{x}{y}}}={21}{e}^{{{7}{x}}}\)
\(\displaystyle{z}_{{{y}{y}}}={0}\)
\(\displaystyle{z}_{{{y}{x}}}={21}{e}^{{{7}{x}}}\)
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