Inverse Trigonometric Functions

Step 1: Applying the chain rule.
We are given the function . To differentiate this, we use the chain rule, which states: where .
Step 2: Differentiating the inner function.
The derivative of is .
Step 3: Final derivative.
Applying the chain rule, we get: which simplifies to: Conclusion:
Thus, the statement is True.

Step 1: Understanding the inverse tangent function.
The principal value of is the angle in the range such that .
Step 2: Applying the inverse tangent to .
We know that: Step 3: Conclusion.
Thus, the principal value of is , corresponding to option (b).

Step 1: Using the identity for inverse trigonometric functions.
We know that: This identity holds for all values of .
Step 2: Conclusion.
Thus, the value of is .