Question

Differentiate the function.

6.\(f\left( x \right) = \ln \left( {{{\sin }^2}x} \right)\)

Short answer

The value of the derivative is \(f'\left( x \right) = 2\cot x\)

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 the given function

\(f\left( x \right) = \ln \left( {{{\sin }^2}x} \right)\)

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

\(\begin{aligned}{c}f\left( x \right)&= \ln {\left( {\sin x} \right)^2}\\f\left( x \right)&= 2\ln \left( {\sin x} \right)\\\frac{{df\left( x \right)}}{{dx}}&= 2\frac{{d\ln \left( {\sin x} \right)}}{{d\sin x}} \times \frac{{d\sin x}}{{dx}}\\f'\left( x \right)&= \frac{2}{{\sin x}} \times \cos x\\f'\left( x \right)&= 2\frac{{\cos x}}{{\sin x}}\\f'\left( x \right)&= 2\cot x\end{aligned}\)

Thus, the value of the derivative is \(f'\left( x \right) =2\cot x\).