This example proves of the derivative of the sine function shown below.

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

Expand the expression using the sum of two angles identity.

Split the fraction in two.

Split the limit in two.

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.

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

Then take both limits.

Substitute the values of the limits into the expression.

Simplify the expression.

This proves the derivative of sine.

The derivative of a functin with respect to a variable returns a function that represents the change in the function with repsect to time.

The sum of two angles identities express the cosine and sine of the sum of two angles in terms of their individual cosine and sine components.

The sine function returns the sine of a number provided in radians. In geometric terms, the function returns the vertical component of the point formed by the angle on the unit circle.