The derivation of `sec^(2)x` w.r.t. `tan^(2)x` is

Options

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

Answer

To find the derivative of \( \sec^2 x \) with respect to \( \tan^2 x \), we can use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the functions
Let \( y = \sec^2 x \) and \( u = \tan^2 x \). We need to find \( \frac{dy}{du} \).

### Step 2: Find \( \frac{dy}{dx} \)
First, we differentiate \( y = \sec^2 x \) with respect to \( x \):
\[ \frac{dy}{dx} = 2 \sec^2 x \tan x \]
This is because the derivative of \( \sec x \) is \( \sec x \tan x \), and applying the chain rule gives us the factor of 2.

### Step 3: Find \( \frac{du}{dx} \)
Next, we differentiate \( u = \tan^2 x \) with respect to \( x \):
\[ \frac{du}{dx} = 2 \tan x \sec^2 x \]
Again, this is due to the chain rule, where the derivative of \( \tan x \) is \( \sec^2 x \).

### Step 4: Apply the chain rule to find \( \frac{dy}{du} \)
Using the chain rule, we have:
\[ \frac{dy}{du} = \frac{dy}{dx} \cdot \frac{dx}{du} \]
We can express \( \frac{dx}{du} \) as:
\[ \frac{dx}{du} = \frac{1}{\frac{du}{dx}} = \frac{1}{2 \tan x \sec^2 x} \]

Now substituting \( \frac{dy}{dx} \) and \( \frac{du}{dx} \):
\[ \frac{dy}{du} = \left(2 \sec^2 x \tan x\right) \cdot \left(\frac{1}{2 \tan x \sec^2 x}\right) \]

### Step 5: Simplify the expression
The \( 2 \) and \( \tan x \sec^2 x \) terms cancel out:
\[ \frac{dy}{du} = 1 \]

### Final Answer
Thus, the derivative of \( \sec^2 x \) with respect to \( \tan^2 x \) is:
\[ \frac{dy}{du} = 1 \]

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