Question

\(\frac{d}{dx} \left( \tan^{-1} \frac{x}{a} \right) =\)

• A. \(\frac{a}{a^2 - x^2}\)
• B. \(\frac{1}{a^2 + x^2}\)
• C. \(\frac{1}{a^2 - x^2}\)
• D. \(\frac{a}{a^2 + x^2}\)

Hint:

The derivative of \(\tan^{-1}\left(\frac{x}{a}\right)\) is a standard formula in calculus. It's highly recommended to memorize this result, \(\frac{a}{a^2+x^2}\), and the similar one for \(\sin^{-1}\left(\frac{x}{a}\right)\) to save time in exams.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Find dy/dx for y=√(sin x+√(sin x+...∞)).

Step 2: Key Formula (Alternate):
Square both sides: y²=sin x+y. Then differentiate implicitly.

Step 3: Detailed Explanation:
2y·dy/dx = cos x + dy/dx → (2y-1)dy/dx = cos x → dy/dx = cos x/(2y-1).

Step 4: Final Answer:
Derivative is cos x/(2y-1).