Differentiation

Step 1: Find the derivative of
The first derivative of is:
Step 2: Solve for Critical Points
Set the derivative equal to zero to find the critical points: Simplifying: Factoring: Thus, the critical points are and .
Step 3: Determine the intervals of increase and decrease
Using a sign chart, test the intervals around the critical points and :
For , (function is increasing)
For , (function is decreasing)
For , (function is increasing)

Step 4: Conclusion
Thus, is decreasing in the interval .

The given function is: To differentiate this, apply the product rule for the first term and use the derivative formula for : - The derivative of is: - The derivative of is: Thus, combining these results:
Step 4: Conclusion. The derivative is , which corresponds to option (b).