The derivative of `tan^(-1)(2x)` w.r.t. x is:

Options

A. `1/(1+4x^2)`
B. `2/(1+4x^2)`
C. `(2)/(sqrt(1-4x^2))`
D. `(-2)/(sqrt(1-4x^2))`

Answer

To find the derivative of \( \tan^{-1}(2x) \) with respect to \( x \), we can follow these steps:

### Step 1: Identify the function
We start with the function:
\[ y = \tan^{-1}(2x) \]

### Step 2: Use the derivative formula for \( \tan^{-1}(x) \)
The derivative of \( \tan^{-1}(u) \) with respect to \( x \) is given by:
\[ \frac{d}{dx}(\tan^{-1}(u)) = \frac{1}{1 + u^2} \cdot \frac{du}{dx} \]
where \( u = 2x \).

### Step 3: Differentiate \( u = 2x \)
Now, we need to find \( \frac{du}{dx} \):
\[ u = 2x \implies \frac{du}{dx} = 2 \]

### Step 4: Substitute \( u \) and \( \frac{du}{dx} \) into the derivative formula
Now we substitute \( u = 2x \) and \( \frac{du}{dx} = 2 \) into the derivative formula:
\[ \frac{dy}{dx} = \frac{1}{1 + (2x)^2} \cdot 2 \]

### Step 5: Simplify the expression
Now we simplify the expression:
\[ \frac{dy}{dx} = \frac{2}{1 + 4x^2} \]

### Final Answer
Thus, the derivative of \( \tan^{-1}(2x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = \frac{2}{1 + 4x^2} \]

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