Derivatives

If , then we need to find .

Take the natural logarithm of both sides:

Now, we differentiate with respect to x:

Using the quotient rule:

Therefore, the correct option is (A) .

Statement 1: We are given that . The derivatives of the logarithmic functions are: and Adding these, we get: Hence, Statement 1 is true.
Statement 2: The derivatives of and are incorrect in the statement. The correct derivatives are: and Hence, Statement 2 is false.

The correct answer is (A) : Statement 1 is true; statement 2 is false.

Given:
We need to find .

Step 1: Simplifying the function
can be written as:

Step 2: Applying chain rule
Using the chain rule, we differentiate . First, differentiate the outer function: So, we get: Next, differentiate with respect to :
Step 3: Combining the results
Now applying the chain rule: Simplifying:
Step 4: Final Answer
The expression simplifies to (since ).

The correct answer is (C).