Question

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

Derivatives
ANSWERED
asked 2021-06-03
Find the derivatives of the functions. \(\displaystyle{r}{\left({x}\right)}={\left({\ln{{\left({x}^{{{2}}}\right)}}}\right)}^{{{2}}}\)

Answers (1)

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
\(\displaystyle{r}'{\left({x}\right)}={2}{\ln{{\left({x}^{{{2}}}\right)}}}{\frac{{{2}}}{{{x}}}}={\frac{{{4}{\ln{{\left({x}^{{{2}}}\right)}}}}}{{{x}}}}\)
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