`"If "y=(tan x)^((tan x)^(tan x))," then find "(dy)/(dx).`

Explanation

To solve the problem \( y = (\tan x)^{(\tan x)^{(\tan x)}} \) and find \( \frac{dy}{dx} \), we will use logarithmic differentiation. Here’s the step-by-step solution:

### Step 1: Take the logarithm of both sides
We start by taking the natural logarithm of both sides to simplify the expression:
\[ \log y = \log \left((\tan x)^{(\tan x)^{(\tan x)}}\right) \]

### Step 2: Apply the logarithmic power rule
Using the property of logarithms, \( \log(a^b) = b \log a \), we can rewrite the right-hand side:
\[ \log y = (\tan x)^{(\tan x)} \cdot \log(\tan x) \]

### Step 3: Take the logarithm again
To further simplify, we take the logarithm of both sides again:
\[ \log(\log y) = \log\left((\tan x)^{(\tan x)} \cdot \log(\tan x)\right) \]

### Step 4: Apply the logarithmic product rule
Using the property \( \log(ab) = \log a + \log b \):
\[ \log(\log y) = \log\left((\tan x)^{(\tan x)}\right) + \log(\log(\tan x)) \]
Now applying the power rule again:
\[ \log(\log y) = (\tan x) \cdot \log(\tan x) + \log(\log(\tan x)) \]

### Step 5: Differentiate both sides with respect to \( x \)
Now we differentiate both sides:
\[ \frac{1}{\log y} \cdot \frac{dy}{dx} = \frac{d}{dx}\left((\tan x) \cdot \log(\tan x)\right) + \frac{d}{dx}(\log(\log(\tan x))) \]

### Step 6: Differentiate the right-hand side using the product rule
Using the product rule on the first term:
\[ \frac{d}{dx}\left((\tan x) \cdot \log(\tan x)\right) = \tan x \cdot \frac{d}{dx}(\log(\tan x)) + \log(\tan x) \cdot \frac{d}{dx}(\tan x) \]
Calculating each derivative:
- \( \frac{d}{dx}(\log(\tan x)) = \frac{1}{\tan x} \cdot \sec^2 x \)
- \( \frac{d}{dx}(\tan x) = \sec^2 x \)

Thus, we have:
\[ \frac{d}{dx}\left((\tan x) \cdot \log(\tan x)\right) = \tan x \cdot \left(\frac{1}{\tan x} \cdot \sec^2 x\right) + \log(\tan x) \cdot \sec^2 x \]
This simplifies to:
\[ \sec^2 x + \log(\tan x) \cdot \sec^2 x \]

### Step 7: Differentiate the second term
Now we differentiate the second term:
\[ \frac{d}{dx}(\log(\log(\tan x))) = \frac{1}{\log(\tan x)} \cdot \frac{1}{\tan x} \cdot \sec^2 x \]

### Step 8: Combine the results
Combining the derivatives, we have:
\[ \frac{1}{\log y} \cdot \frac{dy}{dx} = \sec^2 x + \log(\tan x) \cdot \sec^2 x + \frac{\sec^2 x}{\tan x \cdot \log(\tan x)} \]

### Step 9: Solve for \( \frac{dy}{dx} \)
Multiplying both sides by \( \log y \):
\[ \frac{dy}{dx} = \log y \left(\sec^2 x + \log(\tan x) \cdot \sec^2 x + \frac{\sec^2 x}{\tan x \cdot \log(\tan x)}\right) \]

### Final Expression
Substituting back \( y \):
\[ \frac{dy}{dx} = \log\left((\tan x)^{(\tan x)^{(\tan x)}}\right) \left(\sec^2 x + \log(\tan x) \cdot \sec^2 x + \frac{\sec^2 x}{\tan x \cdot \log(\tan x)}\right) \]