`d/(dx)(secx)` is :

Options

A. `tanx`
B. `tanx.secx`
C. `-cot^(2)x`
D. `cosecx`

Answer

To find the derivative of \( \sec x \) with respect to \( x \), we can use the quotient rule. Here’s a step-by-step solution:

### Step 1: Rewrite \( \sec x \)
We know that:
\[ \sec x = \frac{1}{\cos x} \]

### Step 2: Apply the Quotient Rule
The quotient rule states that if you have a function \( y = \frac{a}{b} \), then the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = \frac{b \frac{da}{dx} - a \frac{db}{dx}}{b^2} \]
In our case, \( a = 1 \) and \( b = \cos x \).

### Step 3: Differentiate \( a \) and \( b \)
- The derivative of \( a = 1 \) is:
\[ \frac{da}{dx} = 0 \]
- The derivative of \( b = \cos x \) is:
\[ \frac{db}{dx} = -\sin x \]

### Step 4: Substitute into the Quotient Rule
Now substituting into the quotient rule:
\[ \frac{d}{dx}(\sec x) = \frac{\cos x \cdot 0 - 1 \cdot (-\sin x)}{\cos^2 x} \]
This simplifies to:
\[ \frac{d}{dx}(\sec x) = \frac{\sin x}{\cos^2 x} \]

### Step 5: Rewrite the Result
We can rewrite \( \frac{\sin x}{\cos^2 x} \) using trigonometric identities:
\[ \frac{\sin x}{\cos^2 x} = \tan x \cdot \sec x \]

### Final Answer
Thus, the derivative of \( \sec x \) is:
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \]

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