Question

Assertion (A): The derivative of the function \( f(x) = x^2 \) with respect to \( x \) is \( 2x \).
Reason (R): The derivative of a power function \( x^n \) is given by the formula \( \frac{d}{dx}(x^n) = nx^{n-1} \).

• A. Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are correct, but Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is correct, but Reason (R) is incorrect.
• D. Assertion (A) is incorrect, but Reason (R) is correct.

Hint:

To differentiate a power function \( x^n \), use the formula \( \frac{d}{dx}(x^n) = nx^{n-1} \).

Answer 1

The Correct Option is A

Step 1: Assertion (A) analysis.
The derivative of \( f(x) = x^2 \) is indeed \( 2x \), according to basic differentiation rules.
Step 2: Reason (R) analysis.
Reason (R) is also correct because the general rule for differentiating \( x^n \) is \( \frac{d}{dx}(x^n) = nx^{n-1} \), and applying this to \( f(x) = x^2 \) results in \( 2x \).
Step 3: Conclusion.
Thus, both Assertion (A) and Reason (R) are correct, and Reason (R) correctly explains Assertion (A). Final Answer:} (A) Both Assertion (A) and Reason (R) are correct, and Reason (R) is the correct explanation of Assertion (A).