Question

# 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)

Multivariable functions

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

2021-03-02
a) $$\displaystyle{g}_{{r}}=\frac{{\partial{g}}}{{\partial{r}}}=\frac{\partial}{{\partial{r}}}{\left[{t}{\ln{{r}}}+{11}{r}{t}^{{7}}-{5}{\left({8}^{{r}}\right)}-{t}{r}\right]}$$
$$\displaystyle{g}_{{r}}=\frac{{t}}{{r}}+{11}{t}^{{7}}-{4}{\left({8}^{{r}}\right)}{\ln{{\left({8}\right)}}}-{t}$$
b) $$\displaystyle\frac{{\partial{g}_{{r}}}}{{\partial{r}}}=\frac{\partial}{{\partial{r}}}={\left[\frac{{t}}{{r}}+{11}{t}^{{7}}-{4}{\left({8}^{{r}}\right)}{\ln{{\left({8}\right)}}}-{t}\right]}$$
$$\displaystyle{g}_{{{r}{r}}}=-\frac{{t}}{{r}^{{2}}}+{0}-{4}{\left({8}^{{r}}\right)}{\left({{\ln{{\left({8}\right)}}}^{{2}}-}{0}\right.}$$
$$\displaystyle{g}_{{{r}{r}}}=-\frac{{t}}{{r}^{{2}}}-{4}{\left({8}^{{r}}\right)}{\left({{\ln{{\left({8}\right)}}}^{{2}}}\right.}$$
c) $$\displaystyle\frac{{\partial{g}_{{r}}}}{{\partial{t}}}=\frac{\partial}{{\partial{t}}}={\left[\frac{{t}}{{r}}+{11}{t}^{{7}}-{4}{\left({8}^{{r}}\right)}{\ln{{\left({8}\right)}}}-{t}\right]}$$
$$\displaystyle{g}_{{{r}{t}}}=\frac{{1}}{{r}}+{11}\cdot{7}{t}^{{6}}-{0}-{1}$$
$$\displaystyle{g}_{{{r}{t}}}=\frac{{1}}{{r}}+{77}{t}^{{6}}-{1}$$
d) $$\displaystyle\frac{{\partial{g}}}{{\partial{t}}}=\frac{\partial}{{\partial{t}}}{\left[{t}{\ln{{\left({r}\right)}}}+{11}{r}{t}^{{7}}-{4}{\left({8}^{{r}}\right)}-{t}{r}\right]}$$
$$\displaystyle{g}_{{t}}={\ln{{\left({r}\right)}}}+{11}{r}\cdot{7}{t}^{{6}}-{0}-{r}$$
$$\displaystyle{g}_{{t}}={\ln{{\left({r}\right)}}}+{77}{r}{t}^{{6}}-{r}$$
e) $$\displaystyle\frac{{\partial{g}_{{t}}}}{{\partial{t}}}=\frac{\partial}{{\partial{t}}}{\left[{\ln{{\left({r}\right)}}}+{77}{r}{t}^{{6}}-{r}\right]}$$
$$\displaystyle{g}_{{\mathtt{=}}}{0}+{77}{r}\cdot{6}{t}^{{5}}-{0}$$
$$\displaystyle{g}_{{\mathtt{=}}}{462}{r}{t}^{{5}}$$