Question

11-18: Find the differential of the function.

17. \(y = \ln \left( {\sin \theta } \right)\)

Short answer

The differential of the function is \(dy = \cot \theta d\theta \).

Step-by-step solution

Definition of differentials

The equation establishes the differential \(dy\)with respect to \(dx\) as shown below:

\(dy = f'\left( x \right)dx\)

Where \(dx\) represent theindependent variable and \(dy\) represent the dependent variable.

Determine the differential of the function

The given function is \(y = f\left( \theta \right) = \ln \left( {\sin \theta } \right)\).

Evaluate the derivative of the function as shown below:

\(\begin{aligned}f'\left( \theta \right) &= \frac{d}{{d\theta }}\ln \left( {\sin \theta } \right)\\ &= \left( {\frac{1}{{\sin \theta }}} \right) \cdot \left( {\cos \theta } \right)\\ &= \cot \theta \end{aligned}\)

Obtain the differential of the function as shown below:

\(\begin{aligned}dy &= f'\left( \theta \right)d\theta \\dy &= \cot \theta d\theta \end{aligned}\)

Thus, the differential of the function is \(dy = \cot \theta d\theta \).