Differentiation

Step 1: Establishing the Functional Equation Since the function follows a repeating pattern, we define: Since the same function appears infinitely, we square both sides: Rearrange:
Step 2: Differentiate Implicitly Differentiating both sides with respect to : Rearranging:
Step 3: Compute Second Derivative Differentiating again and substituting , , we obtain: Thus, the correct answer is:

Step 1: Understanding the Chain Rule Given the function: To differentiate this function, we use the Chain Rule, which states:
Step 2: Evaluating the Function at Since , we can substitute: Therefore, all occurrences of simplify to , so:
Step 3: Computing the Final Derivative Value Given that , we substitute:
Final Answer: Thus, the correct option is: Option (4): 243

Step 1: Checking Continuity at Since , multiplying by : Taking limits, , which equals . So, is continuous.

Step 2: Checking Differentiability at Differentiating, For to exist, is needed to make the second term vanish.