The derivative of `sec^(2)x` with respect to `tan^(2)x` is

Options

A. 1
B. 2
C. `2sec x tan x`
D. `2sec^(2)x tanx`

Explanation

To find the derivative of \( \sec^2 x \) with respect to \( \tan^2 x \), we can use the chain rule. Let's denote:

- \( u = \sec^2 x \)
- \( v = \tan^2 x \)

We want to find \( \frac{du}{dv} \).

### Step 1: Find \( \frac{du}{dx} \)

To find \( \frac{du}{dx} \), we differentiate \( u = \sec^2 x \):

\[ \frac{du}{dx} = 2 \sec^2 x \cdot \tan x \]

### Step 2: Find \( \frac{dv}{dx} \)

Next, we differentiate \( v = \tan^2 x \):

\[ \frac{dv}{dx} = 2 \tan x \cdot \sec^2 x \]

### Step 3: Use the chain rule to find \( \frac{du}{dv} \)

Now, we can use the chain rule to find \( \frac{du}{dv} \):

\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{2 \sec^2 x \tan x}{2 \tan x \sec^2 x} \]

### Step 4: Simplify the expression

The \( 2 \sec^2 x \tan x \) in the numerator and \( 2 \tan x \sec^2 x \) in the denominator cancel out:

\[ \frac{du}{dv} = 1 \]

### Final Answer

Thus, the derivative of \( \sec^2 x \) with respect to \( \tan^2 x \) is:

\[ \frac{du}{dv} = 1 \]

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