Question

# Find the derivatives of the functions. r(x)=(\ln (x^{2}))^{2}

Derivatives
Find the derivatives of the functions. $$\displaystyle{r}{\left({x}\right)}={\left({\ln{{\left({x}^{{{2}}}\right)}}}\right)}^{{{2}}}$$

2021-06-04
Using the chain rule, we get
$$\displaystyle{r}'{\left({x}\right)}={2}{\ln{{\left({x}^{{{2}}}\right)}}}{\left[{\ln{{\left({x}^{{{2}}}\right)}}}\right]}'$$
Using the logarithm property $$\displaystyle{{\ln{{x}}}^{{{2}}}=}{a}{\ln{{x}}}$$, we get
$$\displaystyle{\ln{{\left({x}^{{{2}}}\right)}}}={2}{\ln{{x}}}$$
Thus
$$\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{\ln{{\left({x}^{{{2}}}\right)}}}\right]}={\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left[{2}{\ln{{x}}}\right]}={\frac{{{2}}}{{{x}}}}$$
Combine everything to get
$$\displaystyle{r}'{\left({x}\right)}={2}{\ln{{\left({x}^{{{2}}}\right)}}}{\frac{{{2}}}{{{x}}}}={\frac{{{4}{\ln{{\left({x}^{{{2}}}\right)}}}}}{{{x}}}}$$