Question
Solve this : $$\frac{{{d^n}}}{{d{x^n}}}\left( {\log \,x} \right) = ?$$
A.
$$\frac{{\left( {n - 1} \right)!}}{{{x^n}}}$$
B.
$$\frac{{n!}}{{{x^n}}}$$
C.
$$\frac{{\left( {n - 2} \right)!}}{{{x^n}}}$$
D.
$${\left( { - 1} \right)^{n - 1}}\frac{{\left( {n - 1} \right)!}}{{{x^n}}}$$
Answer :
$${\left( { - 1} \right)^{n - 1}}\frac{{\left( {n - 1} \right)!}}{{{x^n}}}$$
Solution :
Let $$y = \log \,x$$
$$ \Rightarrow {y_1} = \frac{1}{x},\,{y_2} = \frac{{ - 1}}{{{x^2}}},\,{y_3} = \frac{2}{{{x^3}}},.....,\,{y_n} = \frac{{{{\left( { - 1} \right)}^{n - 1}}\left( {n - 1} \right)!}}{{{x^n}}}$$