More: Finding the derivative of the sine and cosine functions

When y(x) = sin(x), then by the method of limits and the difference quotient:

This limit is not as easy to solve as those you have previously done. You must first place the expression into a different form. You do this by using one of the basic addition formulas from trigonometry.

Now you can place the expression above into a form that you can manipulate algebraically.

• In LiveMath, input the expression into its own Prop
  
• Select sin(x+h) and Transform. Choose the second choice.
  
• Mini-Expand the numerator (this gets rid of the parenthesis).
  
• Select everything to the right of, and including, -sin(x). Manipulate with a Collect.
  
  
• Mini-Expand the whole expression.
  

Below, the expression is placed into a more familiar form which you can apply two limit rules (LiveMath does have some manipulation limitations).

The two quotients in this expression are important limits. The first one;

is equal to one (=1). You can verify this by entering, as above, and creating a Table where the variable h is used and is given values from 1 to 0.

When you open the table you will see that, as h goes to 0, the expression goes to 1.

You can solve the second limit in the same manor and you will find that it will go to 0 as h goes to 0.

This then makes the derivative look like the following:

There you have it. A very important derivative, that is quite obvious when looking at a plot of the sine and cosine functions, is verified.

 You can derive the cosine function in the same fashion by using the other addition formula.

where:

Continuing;

Using limit algebra you now have;