Question

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

Derivatives

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

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:
$$\displaystyle{z}_{{\times}}={147}{y}{e}^{{{7}{x}}}$$
$$\displaystyle{z}_{{{x}{y}}}={21}{e}^{{{7}{x}}}$$
$$\displaystyle{z}_{{{y}{y}}}={0}$$
$$\displaystyle{z}_{{{y}{x}}}={21}{e}^{{{7}{x}}}$$