Question

Find the \(n{\rm{th}}\) derivative of each function by calculating the first few derivatives and observing the pattern that occurs.

1. \(f\left( x \right) = {x^n}\)
2. \(f\left( x \right) = \frac{1}{{{x^n}}}\)

Short answer

1. The \(n{\rm{th}}\) derivative of the function is \({f^{\left( n \right)}}\left( x \right) = n!\).
2. The \(n{\rm{th}}\) derivative of the function is \({f^{\left( n \right)}}\left( x \right) = \frac{{{{\left( { - 1} \right)}^n}n!}}{{{x^{n + 1}}}}\).

Step-by-step solution

 The power rule

When \(n\) is any real number, then

\(\frac{d}{{dx}}\left( {{x^n}} \right) = n{x^{x - 1}}\)

(a) Step 2: Find the \(n{\rm{th}}\) derivative of the function

Evaluate the first few derivatives of the function as shown below:

\(\begin{align}f'\left( x \right) &= n{x^{n - 1}}\\f''\left( x \right) &= n\left( {n - 1} \right){x^{n - 2}}\\ \vdots \end{align}\)

Observe the above derivatives pattern to determine the nth derivative of the function as shown below:

\(\begin{align}{f^{\left( n \right)}}\left( x \right) &= n\left( {n - 1} \right)\left( {n - 2} \right) \ldots 2 \cdot 1{x^{n - n}}\\ &= n!\end{align}\)

Thus, the \(n{\rm{th}}\) derivative of the function is \({f^{\left( n \right)}}\left( x \right) = n!\).

(b) Step 3: Find the \(n{\rm{th}}\) derivative of the function

Rewrite the function as the power of \(x\) as shown below:

\(\begin{align}f\left( x \right) &= \frac{1}{x}\\ &= {x^{ - 1}}\end{align}\)

Evaluate the first few derivatives of the function as shown below:

\(\begin{align}f'\left( x \right) &= \left( { - 1} \right){x^{ - 1 - 1}}\\ &= \left( { - 1} \right){x^{ - 2}}\\f''\left( x \right) &= \left( { - 1} \right)\left( { - 2} \right){x^{n - 2}}\\ \vdots \end{align}\)

Observe the above derivatives pattern to determine the nth derivative of the function as shown below:

\(\begin{align}{f^{\left( n \right)}}\left( x \right) &= \left( { - 1} \right)\left( { - 2} \right)\left( { - 3} \right) \ldots \left( { - n} \right){x^{ - \left( {n + 1} \right)}}\\ &= \frac{{{{\left( { - 1} \right)}^n}n!}}{{{x^{n + 1}}}}\end{align}\)

Thus, the \(n{\rm{th}}\) derivative of the function is \({f^{\left( n \right)}}\left( x \right) = \frac{{{{\left( { - 1} \right)}^n}n!}}{{{x^{n + 1}}}}\).