Limits And Derivatives

We need to compute the derivative of the given expression. The limit as of simplifies to . Differentiating this with respect to , we get: Thus, the correct answer is option (B) .

The derivative of is evaluated by first considering the absolute value function. We know that: Thus, for , the derivative is: Therefore, the correct answer is option (B) 4.

We have the function . Using the properties of logarithms, we can simplify the function: Now, we take the derivative of this with respect to : Thus, the derivative is: Therefore, the correct answer is , which corresponds to option (C).

The given function is: Step 1: Apply the logarithmic property
We can simplify the expression using the logarithmic identity , so the expression becomes:
Step 2: Differentiate
Now differentiate each term: The derivative of with respect to is 0 because is a constant. Thus, the derivative is: