Question

f(x) = 3/x(lnx)^2
f(x)' = ?
This one is tricky, can someone explain?

Answer 1

Start by rewriting this in a form that's more friendly for the power rule:
\[\frac{ 3 }{ x \ln (x)^2 } \rightarrow 3x^{-1}\ln(x)^{-2}\]

Answer 2

Then, use the power rule:
Identify f(x) and g(x):
\[f(x) = 3x^{-1} \\ g(x) = \ln(x)^{-2}\]
and now find f'(x) and g'(x):
\[f'(x) = -3x^{-2} \\ g'(x) = (-2\ln(x)^{-3})x^{-1}\]
G'(x) found using the chain rule

Answer 3

\[\int\limits \left(-\frac{3 (\log (x)+2)}{x^2 \log ^3(x)}\right) \, dx=\frac{3}{x \text{Log}[x]^2} \]