Question

The derivative of \(x^x\) with respective to x is

• A. \(x^x(x + \log x)\)
• B. \(x^x(x - \log x)\)
• C. \(x^x(1 - \log x)\)
• D. \(x^x(1 + \log x)\)

Hint:

The derivative of \(x^x\) is a standard result that's good to remember. Whenever you see a function in the form of (variable)(variable), logarithmic differentiation is the method to use.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Find d/dx[tan⁻¹(x/a)].

Step 2: Key Formula (Alternate):
Use standard formula: d/dx[tan⁻¹(x/a)] = a/(a²+x²).

Step 3: Detailed Explanation:
Chain rule: let u=x/a, d/dx[tan⁻¹u] = (1/(1+u²))·(1/a) = a/(a²+x²) after simplification.

Step 4: Final Answer:
Derivative is a/(a²+x²).