Differentiation

Step 1: Differentiate with respect to . Given: Differentiate both sides with respect to : Step 2: Differentiate with respect to . Given: Differentiate both sides with respect to : Step 3: Compute . Using the chain rule: Simplify: Using the identity: and we get: Canceling common terms: Conclusion: The correct answer is .

Concept: Function is decreasing when
Step 1: Differentiate.
Step 2:
Step 3: Sign analysis.

Concept: Marginal revenue = derivative of revenue function.
Step 1: Differentiate.
Step 2:

Concept: Use implicit differentiation.
Step 1: Take log.
Step 2: First derivative.
Step 3: Second derivative.
Step 4:

Concept:
Step 1: Rewrite function.
Step 2:
Step 3:
Step 4:

Concept: For parametric equations:
Step 1: Differentiate.
Step 2:
Step 3: Express in : Substituting gives: