Step 1: Use the self-repeating trick.
The expression under the first root repeats forever: it is $\tan x$ plus the same whole thing again. So the entire expression equals $y$. That lets us write $y = \sqrt{\tan x + y}$.
Step 2: Square both sides.
Squaring removes the outer root and gives a clean equation.
\[ y^2 = \tan x + y \]
Step 3: Get ready to differentiate.
Now differentiate both sides with respect to $x$. We treat $y$ as a function of $x$, so its derivative is $\frac{dy}{dx}$.
Step 4: Differentiate each term.
The left $y^2$ gives $2y\frac{dy}{dx}$. The $\tan x$ gives $\sec^2 x$, and the $y$ gives $\frac{dy}{dx}$.
\[ 2y\frac{dy}{dx} = \sec^2 x + \frac{dy}{dx} \]
Step 5: Collect the $\frac{dy}{dx}$ terms.
Move the $\frac{dy}{dx}$ from the right to the left.
\[ 2y\frac{dy}{dx} - \frac{dy}{dx} = \sec^2 x \]
Step 6: Factor out $\frac{dy}{dx}$.
Take the common factor on the left.
\[ (2y - 1)\frac{dy}{dx} = \sec^2 x \]
\[ \boxed{\sec^2 x} \]