Differentiation

Step 1: Understanding the Concept
The problem requires finding the derivative of a function involving a logarithm with a variable base and then evaluating it at a specific point. The first step is to simplify the function using logarithm properties, particularly the change of base formula.
Step 2: Key Formula or Approach
1. Use the change of base formula for logarithms: . 2. Simplify the resulting expression using properties of logarithms like and . 3. Differentiate the simplified function using the quotient rule: . 4. Substitute into the derivative to find the final value.
Step 3: Detailed Explanation
1. Simplify the function : The function is . First, factor the argument of the logarithm: . Apply the change of base formula to natural logarithm (ln): Using logarithm properties: (The domain ensures that the arguments of the logarithms are well-defined and positive for ). Split the fraction:
2. Differentiate : Using the quotient rule for the fraction:
3. Evaluate : Substitute into the expression for : Since :
4. Match with options: The result is . We can rewrite this using logarithm properties: Using the change of base formula again, .
Step 4: Final Answer
The value of the derivative at is .

Step 1: Check for continuity at .
A function is continuous at if . Here, we need to check if . We use the Squeeze Theorem. We know that for any : Multiplying by (which is non-negative): Since and , by the Squeeze Theorem: Since the limit equals , the function is continuous at .

Step 2: Check for differentiability at .
We use the first principle definition of the derivative:

Step 3: Evaluate the limit for the derivative.
We use the Squeeze Theorem again. Multiplying by : If , . If , . In either case, as , the function is squeezed between two functions that approach 0. Since the limit exists and is finite, the function is differentiable at , and .

Step 1: Calculate the first derivative, }.
First, find the derivatives with respect to the parameter . The first derivative is given by the chain rule: Using half-angle identities, this simplifies to .

Step 2: Calculate the second derivative, .}
We use the formula .

Step 3: Use the given condition to find the value of K.
We are given that at , . Taking the fourth root, (since cosecant of half angle for cycloid is positive). Since , the smallest positive solution is , which gives .

Step 4: Evaluate the required limit.
We need to find . Since the function is continuous at , we can directly substitute the value.