Question

Differentiate the function.

5.\(f\left( x \right) = \sin \left( {\ln x} \right)\)

Short answer

The value of the derivative is \(f'\left( x \right) = \frac{{\cos \left( {\ln x} \right)}}{x}\).

Step-by-step solution

Use the 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

\(f\left( x \right) = \sin \left( {\ln x} \right)\)

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

\(\begin{aligned}{l}\frac{{df\left( x \right)}}{{dx}}&= \frac{{d\sin \left( {\ln x} \right)}}{{dx}}\\f'\left( x \right)&= \frac{{d\sin \left( {\ln x} \right)}}{{d\ln x}} \times \frac{{d\ln x}}{{dx}}\\f'\left( x \right)&= \cos \left( {\ln x} \right)\frac{1}{x}\\f'\left( x \right)&= \frac{{\cos \left( {\ln x} \right)}}{x}\end{aligned}\)

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