Derivative of `sec^(2)(x^(2))` with respect to `x^(2)` is:

Options

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

Answer

To find the derivative of \( \sec^2(x^2) \) with respect to \( x^2 \), we can follow these steps:

### Step 1: Define the Variables
Let:
- \( u = \sec^2(x^2) \)
- \( v = x^2 \)

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

### Step 2: Differentiate \( u \) with respect to \( x \)
Using the chain rule, we first differentiate \( u \) with respect to \( x \):
\[ \frac{du}{dx} = \frac{d}{dx}(\sec^2(x^2)) \]
Using the chain rule:
\[ \frac{du}{dx} = 2\sec^2(x^2) \tan(x^2) \cdot \frac{d}{dx}(x^2) \]
Since \( \frac{d}{dx}(x^2) = 2x \), we have:
\[ \frac{du}{dx} = 2\sec^2(x^2) \tan(x^2) \cdot 2x = 4x \sec^2(x^2) \tan(x^2) \]

### Step 3: Differentiate \( v \) with respect to \( x \)
Now, we differentiate \( v \) with respect to \( x \):
\[ \frac{dv}{dx} = \frac{d}{dx}(x^2) = 2x \]

### Step 4: Apply the Chain Rule
Now we can find \( \frac{du}{dv} \) using the chain rule:
\[ \frac{du}{dv} = \frac{du/dx}{dv/dx} = \frac{4x \sec^2(x^2) \tan(x^2)}{2x} \]
Simplifying this gives:
\[ \frac{du}{dv} = 2 \sec^2(x^2) \tan(x^2) \]

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

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