Proof
We want to find the derivative of . Let . Then, by definition, .
Taking the derivative of both sides with respect to :
Now, we solve for :
Using the identity and knowing :
Thus, the derivative of is:
Explanation
To find the derivative of , we start by letting . This means that , representing the angle whose tangent is .
Next, we differentiate both sides of the equation with respect to . The derivative of with respect to is simply .
On the right side, the derivative of with respect to is , and by the chain rule, we multiply by , which gives us .
Setting the derivatives equal, we get:
We then solve for by dividing both sides by :
We use the trigonometric identity . Since (from our earlier definition), we substitute for :
Thus, the expression for the derivative simplifies to:
Q.E.D.