Derivative of Sine from First Principles

Example:

Prove, from first principles, that the derivative of is .

You may assume that as and .

Note: The assumptions above can be derived from the small angle approximations.

Step 1: Write the limit out clearly.

Step 2: Expand using the addition formula.

Step 3: Now, we collect the like terms and factorise. In this case, it will be .

First, let’s write the two terms containing next to each other.

Step 4: We will split the fraction so that the two terms in the numerator are individually written over .

Step 5: Now, we can split the limit and evaluate each limit function separately, just like how in integration, we integrate term by term.

Step 6: Since the limit depends on , any function of is considered a constant, which means we can take it out of the limit. This is like how we pull out constants when computing integrals and summations.

Step 7: Using the given assumptions, we can evaluate the limits.

Therefore, .