Question

\(\frac{d}{dx}(\sec x) =\)

• A. Cos x
• B. -Cosec²x
• C. Sec x . Tan x
• D. -Cosec x . Cot x

Hint:

Derivatives of trig functions to remember:
\(\frac{d}{dx}(\sin x) = \cos x\)
\(\frac{d}{dx}(\cos x) = -\sin x\)
\(\frac{d}{dx}(\tan x) = \sec^2 x\)
\(\frac{d}{dx}(\sec x) = \sec x \tan x\)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This question tests knowledge of trigonometric differentiation formulas.

Step 2: Key Formula or Approach:
The derivative of sec x is:
\[ \frac{d}{dx}(\sec x) = \sec x \cdot \tan x \]

Step 3: Detailed Explanation:
We can derive this using the quotient rule:
\[ \sec x = \frac{1}{\cos x} \] \[ \frac{d}{dx}\left(\frac{1}{\cos x}\right) = \frac{0 \cdot \cos x - 1 \cdot (-\sin x)}{\cos^2 x} = \frac{\sin x}{\cos^2 x} \] \[ = \frac{1}{\cos x} \cdot \frac{\sin x}{\cos x} = \sec x \cdot \tan x \] Other derivatives:
- \(\frac{d}{dx}(\sin x) = \cos x\) (A)
- \(\frac{d}{dx}(\cot x) = -\csc^2 x\) (B)
- \(\frac{d}{dx}(\csc x) = -\csc x \cdot \cot x\) (D)
Thus, the derivative of sec x is sec x tan x.

Step 4: Final Answer:
Thus, \(\frac{d}{dx}(\sec x) = \sec x \cdot \tan x\).
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