Question

11-18: Find the differential of the function

14. \(y = {\theta ^2}\sin 2\theta \)

Short answer

The differential of the function is \(dy = 2\theta \left( {\theta \cos 2\theta + \sin 2\theta } \right)d\theta \).

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 the independent variableand \(dy\) represent the dependent variable.

Determine the differential of the function

The function is \(y = f\left( \theta \right) = {\theta ^2}\sin 2\theta \).

Evaluate the derivative of the function as shown below:

\(\begin{aligned}f'\left( \theta \right) &= \frac{d}{{d\theta }}\left( {{\theta ^2}\sin 2\theta } \right)\\ &= {\theta ^2}\left( {\cos 2\theta } \right)\left( 2 \right) + \left( {\sin 2\theta } \right)\left( {2\theta } \right)\\ &= 2\theta \left( {\theta \cos 2\theta + \sin 2\theta } \right)\end{aligned}\)

Obtain the differential of the function is shown below:

\(\begin{aligned}dy &= f'\left( \theta \right)d\theta \\dy &= 2\theta \left( {\theta \cos 2\theta + \sin 2\theta } \right)d\theta \end{aligned}\)

Thus, the differential of the function is \(dy = 2\theta \left( {\theta \cos 2\theta + \sin 2\theta } \right)d\theta \).