Logarithmic Differentiation

For the function to be differentiable at , it must be continuous and have matching derivatives at from both sides.
Step 1: Continuity at }
The function must be continuous at .
Thus, the left-hand limit at should be equal to the right-hand limit at .
For , , so: For , . Substituting : For continuity, these must be equal: Thus, we get the equation: \textbf{Step 2: Differentiability at } For the function to be differentiable at , the left-hand derivative should equal the right-hand derivative. For , , so: At : For , , so: For differentiability, we require: So, .
Step 3: Solving for } Substitute into Equation 1: Thus, .

The correct option is (A) :

We are given the function:
Step 1: Simplify the given expression Using the property of logarithms , we can simplify the expression for :
Step 2: Differentiate with respect to Now, we differentiate using the chain rule: Since , we get: Step 3: Simplify the final expression Thus, the derivative is:

The correct option is (B) :