The derivative of the 80386DX processor is the 80386SX processor.

Explanation:

1. Introduction:
The 80386DX processor was a 32-bit microprocessor released by Intel in 1985. It was a significant improvement over its predecessor, the 80286, offering enhanced performance and capabilities. The 80386DX was the first processor in the 80386 family, which also included other derivatives.

2. Derivative Processors:
Derivative processors are variations or modified versions of an existing processor. They are usually created to cater to different market segments or to offer a lower-cost alternative. In the case of the 80386DX, Intel introduced the 80386SX as a derivative.

3. Comparison between 80386DX and 80386SX:
The 80386DX and 80386SX processors were similar in many ways, but there were some key differences:

- Bus Width: The 80386DX had a 32-bit external data bus and a 32-bit internal data bus, whereas the 80386SX had a 16-bit external data bus and a 32-bit internal data bus.
- Performance: The 80386DX offered better performance due to its wider data bus, allowing faster data transfer. The 80386SX, on the other hand, had reduced performance due to its narrower external data bus.
- Cost: The 80386SX was designed to be a more affordable option compared to the 80386DX. By reducing the external data bus width, Intel was able to manufacture the 80386SX at a lower cost.

4. Conclusion:
In conclusion, the derivative of the 80386DX processor is the 80386SX processor. The 80386SX was a modified version of the 80386DX, offering a narrower external data bus and lower cost. Despite its reduced performance, the 80386SX provided an alternative for those who needed a more budget-friendly option.