Question

Which of the following functions has derivative equal to \( \cos x \)?

• A. \( \sin x \)
• B. \( -\sin x \)
• C. \( \tan x \)
• D. \( \sec x \)

Hint:

Be very careful with negative signs in trigonometric derivatives and integrals.
The derivative of \( \sin x \) is \( \cos x \), but the integral of \( \sin x \) is \( -\cos x \).
Memorizing these pairs in a tabular format prevents silly mistakes under exam pressure.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to identify which of the given trigonometric functions, when differentiated with respect to \( x \), results in \( \cos x \).
This is a fundamental question testing the basic derivatives of trigonometric functions.

Step 2: Key Formula or Approach:
We will recall and list the standard derivatives of common trigonometric functions:
1. \( \frac{d}{dx}(\sin x) = \cos x \)
2. \( \frac{d}{dx}(\cos x) = -\sin x \)
3. \( \frac{d}{dx}(\tan x) = \sec^2 x \)
4. \( \frac{d}{dx}(\sec x) = \sec x \tan x \)

Step 3: Detailed Explanation:
Let us test each option by differentiating it with respect to \( x \):
- For Option (A):
\[ \frac{d}{dx}(\sin x) = \cos x \]
This matches the requirement perfectly.
- For Option (B):
\[ \frac{d}{dx}(-\sin x) = -\frac{d}{dx}(\sin x) = -\cos x \]
This has a negative sign, so it is incorrect.
- For Option (C):
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]
This is not \( \cos x \), so it is incorrect.
- For Option (D):
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \]
This is not \( \cos x \), so it is incorrect.
Thus, the only function whose derivative is exactly \( \cos x \) is \( \sin x \).

Step 4: Final Answer:
The correct function is \( \sin x \), which corresponds to Option (A).