The Differentials: Example 4

Problem:

Find d\left(cos\:\theta \:sin\:\theta \right).

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SOLUTION:

We use the product rule to solve this one. So, we have

=cos{\theta}\ d\left(sin{\theta}\right)+sin{\theta}d\left(cos{\theta}\right)

=cos{\theta}cos{\theta}d\theta+sin{\theta}\left(-sin{\theta}\right)d\theta

=\left({cos}^2{\theta}-{sin}^2{\theta}\right)d\theta

=cos{\left(2\theta\right)}d\theta