Proof
Explanation
The proof begins by stating the definition of the derivative of a real function at a point. In this case, it’s the derivative of with respect to , which is the limit as approaches of .
The next step uses the trigonometric identity for the cosine of a sum: . Here, is and is . Applying this identity to , we get: .
The numerator is then rearranged by separating the terms involving and . Specifically, is factored out from the terms involving it, and we write the expression as . The denominator remains unchanged.
The limit is split into two parts using the sum rule for limits. This rule states that the limit of a sum is equal to the sum of the limits, provided both limits exist. So we now have two limits: one for and another for .
We can evaluate each of these limits separately. The limit of as approaches is equal to (this is a standard limit). The limit of as approaches is equal to (this is another standard limit). When we multiply these limits by and respectively, we get and .
Adding these together as per the sum rule for limits, we get , which simplifies to .
QED: Therefore, the derivative of with respect to is .