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

Explanation

To find the derivative \( \frac{dy}{dx} \) for the function \( y = (\tan x)^{(\tan x)^{(\tan x)}} \), we can follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the natural logarithm of both sides to simplify the expression:

\[ \ln y = \ln \left( (\tan x)^{(\tan x)^{(\tan x)}} \right) \]

### Step 2: Apply the logarithmic identity
Using the property of logarithms, \( \ln(a^b) = b \ln a \), we can rewrite the right-hand side:

\[ \ln y = (\tan x)^{(\tan x)} \cdot \ln(\tan x) \]

### Step 3: Take the logarithm again
Next, we take the logarithm of the right-hand side again:

\[ \ln(\ln y) = \ln\left((\tan x)^{(\tan x)} \cdot \ln(\tan x)\right) \]

### Step 4: Expand using logarithmic properties
Using the property \( \ln(m \cdot n) = \ln m + \ln n \):

\[ \ln(\ln y) = \ln\left((\tan x)^{(\tan x)}\right) + \ln(\ln(\tan x)) \]

Now applying the logarithmic identity again:

\[ \ln(\ln y) = (\tan x) \ln(\tan x) + \ln(\ln(\tan x)) \]

### Step 5: Differentiate both sides
Now we differentiate both sides with respect to \( x \):

Using implicit differentiation on the left side:

\[ \frac{1}{\ln y} \cdot \frac{1}{y} \cdot \frac{dy}{dx} = \frac{d}{dx}\left((\tan x) \ln(\tan x) + \ln(\ln(\tan x))\right) \]

### Step 6: Differentiate the right side
Now we differentiate the right side using the product rule and chain rule:

1. For \( (\tan x) \ln(\tan x) \):
- Differentiate using the product rule:
\[ \frac{d}{dx}[(\tan x) \ln(\tan x)] = \sec^2 x \ln(\tan x) + \tan x \cdot \frac{1}{\tan x} \sec^2 x = \sec^2 x \ln(\tan x) + \sec^2 x \]

2. For \( \ln(\ln(\tan x)) \):
- Using the chain rule:
\[ \frac{d}{dx}[\ln(\ln(\tan x))] = \frac{1}{\ln(\tan x)} \cdot \frac{1}{\tan x} \sec^2 x \]

### Step 7: Combine the derivatives
Combining these results:

\[ \frac{1}{\ln y} \cdot \frac{1}{y} \cdot \frac{dy}{dx} = \sec^2 x \ln(\tan x) + \sec^2 x + \frac{\sec^2 x}{\tan x \ln(\tan x)} \]

### Step 8: Solve for \( \frac{dy}{dx} \)
Now, isolate \( \frac{dy}{dx} \):

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

### Step 9: Substitute back for \( y \)
Finally, substitute back \( y = (\tan x)^{(\tan x)^{(\tan x)}} \):

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

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:

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