Question

Find a formula for \({f^{\left( n \right)}}\left( x \right)\) if \(f\left( x \right) = {\bf{ln}}\left( {x - {\bf{1}}} \right)\).

Short answer

The expression for \({f^{\left( n \right)}}\left( x \right)\) is \({\left( { - 1} \right)^{n - 1}}\frac{{\left( {n - 1} \right)!}}{{{{\left( {x - 1} \right)}^n}}}\).

Step-by-step solution

Differentiate the function \(f\left( x \right)\)

Thedifferentiation of \(f\left( x \right)\) with respect to \(x\) is:

\(\begin{aligned}f'\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\ln \left( {x - 1} \right)} \right)\\ &= \frac{1}{{x - 1}}\end{aligned}\)

Find the second derivative of \(f\left( x \right)\)

Differentiate the equation \(f'\left( x \right) = \frac{1}{{x - 1}}\).

\(\begin{aligned}f''\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{1}{{x - 1}}} \right)\\ &= - {\left( {x - 1} \right)^{ - 2}}\\ &= - \frac{1}{{{{\left( {x - 1} \right)}^2}}}\end{aligned}\)

Find the third derivative of the \(f\left( x \right)\)

Differentiate the equation \(f''\left( x \right) = - \frac{1}{{{{\left( {x - 1} \right)}^2}}}\).

\(\begin{aligned}f'''\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( { - \frac{1}{{{{\left( {x - 1} \right)}^2}}}} \right)\\ &= - \left( { - 2{{\left( {x - 1} \right)}^{ - 3}}} \right)\\ &= \frac{2}{{{{\left( {x - 1} \right)}^3}}}\end{aligned}\)

Find the fourth derivative of the \(f\left( x \right)\)

Differentiate the equation \(f'''\left( x \right) = \frac{2}{{{{\left( {x - 1} \right)}^3}}}\).

\(\begin{aligned}{f^{\left( 4 \right)}}\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\frac{2}{{{{\left( {x - 1} \right)}^3}}}} \right)\\ &= 2\left( { - 3} \right){\left( {x - 1} \right)^{ - 4}}\\ &= - \frac{6}{{{{\left( {x - 1} \right)}^4}}}\end{aligned}\)

Find the expression for \({n^{th}}\) derivative of the \(f\left( x \right)\)

From the first four derivatives of \(f\left( x \right)\), the \({n^{{\rm{th}}}}\) derivative of \(f\left( x \right)\) is:

\(\begin{aligned}{f^{\left( n \right)}}\left( x \right) &= {\left( { - 1} \right)^{n - 1}} \cdot 2 \cdot 3 \cdot 4 \cdot \cdot \cdot \cdot \left( {n - 1} \right){\left( {x - 1} \right)^{ - n}}\\ &= {\left( { - 1} \right)^{n - 1}}\frac{{\left( {n - 1} \right)!}}{{{{\left( {x - 1} \right)}^n}}}\end{aligned}\)

Thus, the expression for \({f^{\left( n \right)}}\left( x \right)\) is \({\left( { - 1} \right)^{n - 1}}\frac{{\left( {n - 1} \right)!}}{{{{\left( {x - 1} \right)}^n}}}\).