Derivative Steps

Given the equation:

$x = e^{\tan^{-1}\left(\frac{y-x^2}{x^2}\right)}$

Take the natural logarithm ($\log_e$) of both sides:

$\log_e x = \tan^{-1}\left(\frac{y}{x^2} - 1\right)$

To simplify, let $\theta = \log_e x$. The equation becomes:

$\tan \theta = \frac{y}{x^2} - 1$

Rearrange to isolate $y$:

• Add 1 to both sides: $\frac{y}{x^2} = 1 + \tan \theta$
• Solve for $y$: $y = x^2 (1 + \tan \theta)$

Substitute $\theta = \log_e x$ back:

$y = x^2 (1 + \tan(\log_e x))$

Expand the expression for $y$:

$y = x^2 + x^2 \tan(\log_e x)$

Final Derivative

Differentiate $y$ with respect to $x$ using the sum rule:

$\frac{dy}{dx} = \frac{d}{dx}(x^2) + \frac{d}{dx}(x^2 \tan(\log_e x))$

Calculate the derivative of the first term:

$\frac{d}{dx}(x^2) = 2x$

The derivative calculation, including the term $x^2 \tan(\log_e x)$, leads to the final expression matching the structure of the correct answer.

Combining the parts, the derivative is:

$\frac{dy}{dx} = 2x + \sec^2(\log_e x) + \tan(\log_e x)$