Question

Differentiate the function.

4. \(f\left( x \right) = x ln x - x\)

Short answer

The value of the derivative is \(f\left( x \right) = \ln x\).

Step-by-step solution

Use the derivative of logarithmic function

Rule 2: The derivative of \(\ln x\) is,

\(\frac{d}{{dx}}\left( {\ln x} \right) = \frac{1}{x}\)

Evaluate the derivative of given function

\(f\left( x \right) = x\ln x - x\)

Differentiate \(f\left( x \right)\)with respect \(x\),

\(\begin{aligned}{}\frac{{df\left( x \right)}}{{dx}} &= \frac{{d\left( {x\ln x - x} \right)}}{{dx}}\\f'\left( x \right) &= \frac{{dx\ln x}}{{dx}} - \frac{{dx}}{{dx}}\\f'\left( x \right) &= \left( {x.\frac{{d\ln x}}{{dx}} + \ln x\frac{{dx}}{x}} \right) - 1\\f'\left( x \right) &= \left( {x \times \frac{1}{x} + \left( {\ln x} \right).1} \right) - 1\\f'\left( x \right) &= 1 + \ln x - 1\\f'\left( x \right) &= \ln x\end{aligned}\)

Thus, the value of the derivative is \(f'\left( x \right) = \ln x\).