Switching Circuits

Step 1: Apply the chain rule.
We are tasked with differentiating . Using the chain rule: Simplifying:

Step 2: Substitute the given values.
At , we have , , , and . Substituting these values into the formula:

Step 3: Conclude.
Thus, the correct answer is .

Final Answer:

Step 1: Chain rule proof.
By the chain rule, if is a function of and is also a function of , then: This follows from the fact that and are the rates of change of and with respect to , and their ratio gives the rate of change of with respect to .

Step 2: Find the derivative of with respect to .
We want to find of . First, apply the chain rule: The derivative of with respect to is , and the derivative of with respect to is . Thus, the result is:

Final Answer: The derivative of with respect to is:

Step 1: Understand the circuit.
The given circuit consists of two switches and , with their complements and also involved. The circuit configuration suggests a combination of AND, OR, and NOT logic gates.

Step 2: Assign logic variables.
Let the logic variables be for switch 1, for switch 2, and for the output.

Step 3: Determine the symbolic expression.
Based on the circuit, the expression can be written as: Where: - represents the AND operation between and . - represents the AND operation between and . - The OR operation is denoted by the plus sign .

Step 4: Construct the switching table.
The switching table for the above logic expression is as follows:

Step 5: Interpret the results.
- When both switches are off ( ), the output is off ( ). - When and , the output is on ( ). - When and , the output is on ( ). - When both switches are on ( ), the output is off ( ).

Final Answer: