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The problem asks to differentiate the function:

\[
y = \sqrt{\tan x \sqrt{\tan x \sqrt{\tan x + \dots}}}
\]

with respect to \(x\).

### Let's approach it step by step:

This is a nested radical function that involves an infinite recursion of square roots. Let's define a general expression for simplicity:

\[
y = \sqrt{\tan x \cdot y}
\]

### Step 1: Square both sides

Squaring both sides removes the outer square root:

\[
y^2 = \tan x \cdot y
\]

### Step 2: Rearrange

Rearrange the equation to express it in a form suitable for implicit differentiation:

\[
y^2 - y \cdot \tan x = 0
\]

### Step 3: Differentiate implicitly

Now, differentiate both sides with respect to \(x\). Use the product rule on \(y \cdot \tan x\) and the chain rule for \(y^2\):

\[
2y \frac{dy}{dx} - \left(\frac{dy}{dx} \cdot \tan x + y \cdot \sec^2 x \right) = 0
\]

### Step 4: Solve for \(\frac{dy}{dx}\)

Collect the terms involving \(\frac{dy}{dx}\):

\[
2y \frac{dy}{dx} - \frac{dy}{dx} \cdot \tan x = y \cdot \sec^2 x
\]

Factor out \(\frac{dy}{dx}\) on the left-hand side:

\[
\frac{dy}{dx} \left( 2y - \tan x \right) = y \cdot \sec^2 x
\]

Finally, solve for \(\frac{dy}{dx}\):

\[
\frac{dy}{dx} = \frac{y \cdot \sec^2 x}{2y - \tan x}
\]

### Step 5: Substitute back for \(y\)

Recall that \(y = \sqrt{\tan x \cdot y}\), but the expression is recursive. Therefore, \(\frac{dy}{dx}\) as derived here provides the general form for the derivative.

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Let me know if you would like further details or clarifications! Here are five related questions:

1. How can implicit differentiation be applied to more complex equations involving multiple variables?
2. What is the significance of recursive expressions in calculus?
3. How does one handle differentiation in nested square roots without simplifying?
4. Can this approach be generalized to other transcendental functions, such as sine or cosine?
5. What is the interpretation of this differentiation problem in terms of infinite series?

### Tip:
When differentiating recursive or implicit functions, always check if there is a way to simplify the recursion, either by substitution or by squaring both sides to eliminate nested radicals.

Differentiate the following with respect to x: y = √(tan x √(tan x √(tan x + ...)))

Learn how to differentiate the recursive radical function y = √(tan x √(tan x √(tan x + ...))) using implicit differentiation and the chain rule. This calculus problem covers advanced differentiation techniques suitable for students in Grades 11-12 and college level.

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