Check my work(derivatives):

let F(x) = arctan(3lnx). Find f'(e)

d(3lnx) * d(arctan(3lnx))
3/x * 1/(1 +(3lnx)^2)= 3/x(1+ 9ln^2 x)
f(e) = 3/e(1+ 9ln^2 e) = 3/10e

Question

Check my work(derivatives):

let F(x) = arctan(3lnx).  Find f'(e)

d(3lnx) * d(arctan(3lnx))
3/x  *  1/(1 +(3lnx)^2)= 3/x(1+ 9ln^2 x)
f(e) = 3/e(1+ 9ln^2 e) = 3/10e

anonymous · Answer

What's f?  You specified F

anonymous · Answer

sorry, that's a technical typo.

anonymous · Answer

I tried implicit differentiation$$
\begin{array}{rcl}

f(x) &=& \tan^{-1}(3\ln(x)) \
\tan(f(x)) &=& 3\ln(x) \
\frac{d}{dx}\tan(f(x)) &=& \frac{d}{dx} 3\ln(x) \
\sec^2(f(x))f'(x) &=& \frac{3}{x} \
\sec^2(\tan^{-1}(3\ln(x)))f'(x) &=& \frac{3}{x} \
f'(x) &=& \frac{3\cos^2(\tan^{-1}(3\ln(x)))}{x} \

\end{array}

$$Then plug in $e$ $$
\begin{split}
f'(e) &= \frac{3\cos^2(\tan^{-1}(3\ln(e)))}{e} \
&= \frac{3\cos^2(\tan^{-1}(3))}{e} \
&= \frac{3}{e\sqrt{1^2+3^2}^2} \
&=\frac{3}{10e }


\end{split}
$$