If `x = exp {tan^(-1)((y-x^2)/(x^2))}` then dy/dx equals

Options

A. `2x[1+tan(logx)]+x sec^2(logx)`
B. `x[1+tan(logx)]+x sec^2(logx)`
C. `2x[1+tan(logx)]+x^2 sec^2(logx)`
D. `2x[1+tan(logx)]+ sec^2(logx)`

Explanation

To solve the equation \( x = e^{\tan^{-1}\left(\frac{y - x^2}{x^2}\right)} \) and find \( \frac{dy}{dx} \), we will follow these steps:

### Step 1: Take the natural logarithm of both sides
We start with the equation:
\[ x = e^{\tan^{-1}\left(\frac{y - x^2}{x^2}\right)} \]
Taking the natural logarithm (ln) of both sides gives:
\[ \ln x = \tan^{-1}\left(\frac{y - x^2}{x^2}\right) \]

### Step 2: Apply the tangent function
To eliminate the arctangent, we apply the tangent function to both sides:
\[ \tan(\ln x) = \frac{y - x^2}{x^2} \]

### Step 3: Rearranging the equation
Rearranging the equation to express \( y \):
\[ y - x^2 = x^2 \tan(\ln x) \]
\[ y = x^2 + x^2 \tan(\ln x) \]

### Step 4: Differentiate both sides with respect to \( x \)
Now, we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^2) + \frac{d}{dx}(x^2 \tan(\ln x)) \]

### Step 5: Differentiate the first term
The derivative of \( x^2 \) is:
\[ \frac{d}{dx}(x^2) = 2x \]

### Step 6: Differentiate the second term using the product rule
For the term \( x^2 \tan(\ln x) \), we use the product rule:
\[ \frac{d}{dx}(x^2 \tan(\ln x)) = \frac{d}{dx}(x^2) \cdot \tan(\ln x) + x^2 \cdot \frac{d}{dx}(\tan(\ln x)) \]
Calculating the first part:
\[ \frac{d}{dx}(x^2) \cdot \tan(\ln x) = 2x \tan(\ln x) \]
Now for the second part, we apply the chain rule:
\[ \frac{d}{dx}(\tan(\ln x)) = \sec^2(\ln x) \cdot \frac{d}{dx}(\ln x) = \sec^2(\ln x) \cdot \frac{1}{x} \]
Thus,
\[ x^2 \cdot \frac{d}{dx}(\tan(\ln x)) = x^2 \cdot \sec^2(\ln x) \cdot \frac{1}{x} = x \sec^2(\ln x) \]

### Step 7: Combine the derivatives
Now, we combine the derivatives:
\[ \frac{dy}{dx} = 2x + 2x \tan(\ln x) + x \sec^2(\ln x) \]

### Final Expression
Thus, the final expression for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = 2x(1 + \tan(\ln x)) + x \sec^2(\ln x) \]