10.11 Derivatives of trigonometric functions
10.11.1 Derivative of
What is the derivative of ? First, we can use the definition of the limit and a little algebra.
We want to rewrite this in terms of the two limits we found in the previous section!
10.11.2 Derivative of inverse trig functions
10.11.3 Derivative of
This requires a \saysmart idea here, which is to relate the derivative of , which we don’t know, to the derivative of , which we do know. We can start with this formula, which exploits the property that is the inverse function of (and vice versa),
Then, we can differentiate both sides (using the chain rule), from which we can deduce that
Then, we can use a substitution (another clever idea), in this case by setting we then can deduce that
Or, equivalently (as renaming the variable has no effect on the equation),
10.11.4 Derivative of
The method for finding the derivative of is pretty much the same as the method for finding the derivative of (as in Section 10.11.3).
We start with the identity
which we then differentiate. The result of this is that
which we can simplify by computing the parts we know how to compute which allows us to establish the equation
Therefore, we know that
Now, as when differentiating , we can substitute. In this case we will use rather than , and then write that
This gives us the answer, and if you prefer the variable name to , then we can just rename it, which means that
10.11.5 Derivative of
We start with the definition of , which is that
Then, we can substitute , using which
This is almost what we would like, except that we have a random floating around. This can be removed by noting that , or , i.e.