To find the derivative of , we start by letting:
This means that is the angle whose cosine is . In other words, we have:
Step 1: Differentiate both sides
We differentiate both sides of this equation with respect to . The left side simply becomes . For the right side, using the chain rule:
So, the equation becomes:
Step 2: Solve for
Rearranging the equation to solve for :
Step 3: Express in terms of
Using the Pythagorean identity:
We can express as:
Step 4: Substitute back into the derivative
Substituting this back into our expression for :
Thus, the derivative of is:
Q.E.D.