Logarithmic Differentiation

Step 1: Find the derivative of

Step 2: Solve
Factorize: The critical points are and . Analyze the sign of in intervals:
Step 3: Determine intervals
on .
Conclusion: The function is strictly decreasing on .

Step 1: Take logarithms of both sides
The given equation is: Taking the natural logarithm on both sides, we get:
Step 2: Simplify using logarithmic properties
Using the properties of logarithms: Rearranging the terms to express : Thus, we have:
Step 3: Differentiate with respect to
Differentiate both sides of with respect to using the quotient rule:
Step 4: Simplify the derivative
Simplify the numerator: This reduces to:
Conclusion: The derivative is:

Step 1: Differentiate using the chain rule
We have:
Step 2: Simplify derivatives
Substitute back:
Conclusion: The derivative is .

Step 1: Simplify the expression inside
The given expression is: Using trigonometric identities, rewrite:
Step 2: Transform into a single tangent function
Substituting and :
Step 3: Simplify using
Since , we simplify:
Conclusion: The simplest form is .

Step 1: Condition for continuity
For continuity at , .

Step 2: Evaluate the limit
For : Multiply numerator and denominator by :
Step 3: Substitute

Step 4: Set
Thus, , which matches option (B).

Step 1: {Take logarithms of both sides}
The given equation is: Taking the natural logarithm on both sides, we get: Step 2: {Simplify using logarithmic properties}
Using the properties of logarithms: Rearranging the terms to express : Thus, we have: Step 3: {Differentiate with respect to }
Differentiate both sides of with respect to using the quotient rule: Step 4: {Simplify the derivative}
Simplify the numerator: This reduces to: Conclusion: The derivative is proved to be: