 # Write first and second partial derivativesg(r,t)=t lnr+11rt^7-5(8^r)-tra)g_rb)g_(rr)c)g_(rt)d)g_te)g_(tt) glasskerfu 2021-03-01 Answered

Write first and second partial derivatives
$g\left(r,t\right)=t\mathrm{ln}r+11r{t}^{7}-5\left({8}^{r}\right)-tr$
a)${g}_{r}$
b)${g}_{rr}$
c)${g}_{rt}$
d)${g}_{t}$
e)${g}_{t}$

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a) ${g}_{r}=\frac{\partial g}{\partial r}=\frac{\partial }{\partial r}\left[t\mathrm{ln}r+11r{t}^{7}-5\left({8}^{r}\right)-tr\right]$
${g}_{r}=\frac{t}{r}+11{t}^{7}-4\left({8}^{r}\right)\mathrm{ln}\left(8\right)-t$
b) $\frac{\partial {g}_{r}}{\partial r}=\frac{\partial }{\partial r}=\left[\frac{t}{r}+11{t}^{7}-4\left({8}^{r}\right)\mathrm{ln}\left(8\right)-t\right]$
${g}_{rr}=-\frac{t}{{r}^{2}}+0-4\left({8}^{r}\right)\left({\mathrm{ln}\left(8\right)}^{2}-0$
${g}_{rr}=-\frac{t}{{r}^{2}}-4\left({8}^{r}\right)\left({\mathrm{ln}\left(8\right)}^{2}$
c) $\frac{\partial {g}_{r}}{\partial t}=\frac{\partial }{\partial t}=\left[\frac{t}{r}+11{t}^{7}-4\left({8}^{r}\right)\mathrm{ln}\left(8\right)-t\right]$
${g}_{rt}=\frac{1}{r}+11\cdot 7{t}^{6}-0-1$
${g}_{rt}=\frac{1}{r}+77{t}^{6}-1$
d) $\frac{\partial g}{\partial t}=\frac{\partial }{\partial t}\left[t\mathrm{ln}\left(r\right)+11r{t}^{7}-4\left({8}^{r}\right)-tr\right]$
${g}_{t}=\mathrm{ln}\left(r\right)+11r\cdot 7{t}^{6}-0-r$
${g}_{t}=\mathrm{ln}\left(r\right)+77r{t}^{6}-r$
e) $\frac{\partial {g}_{t}}{\partial t}=\frac{\partial }{\partial t}\left[\mathrm{ln}\left(r\right)+77r{t}^{6}-r\right]$
${g}_{\mathtt{=}}0+77r\cdot 6{t}^{5}-0$
${g}_{\mathtt{=}}462r{t}^{5}$