Prove the Derivative of Sine

This example calculates the derivative of the sine function using the limit definition of the derivative and some trig identities.

1. Start by applying the limit definition of the derivative to the sine function.

2. Expand the expression using the sum of two angles identities.

3. Split the fraction in two.

4. Split the limit in two.

5. Factor the expression which is not related to out of the first limit. Factor out which is not related to out of the second limit.

6. Multiply the second limit by in the form to flip signs.

7. Then take both limits.

   Substitute the values of the limits into the expression.

8. Simplify the expression and this shows the derivative of sine is cosine.