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

Explanation

To find the derivative of the function \( y = (\tan x)^{(\tan x)^{(\tan x)}} \), we will follow these steps:

### Step 1: Take the natural 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, we can bring down the exponent:
\[ \log y = (\tan x)^{(\tan x)} \cdot \log(\tan x) \]

### Step 3: Take the logarithm again
Now, we take the logarithm of both sides again:
\[ \log(\log y) = \log \left( (\tan x)^{(\tan x)} \cdot \log(\tan x) \right) \]
Using the property of logarithms that states \( \log(ab) = \log a + \log b \):
\[ \log(\log y) = \log((\tan x)^{(\tan x)}) + \log(\log(\tan x)) \]

### Step 4: Apply the logarithmic power rule again
We can simplify further:
\[ \log(\log y) = (\tan x) \cdot \log(\tan x) + \log(\log(\tan x)) \]

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

### Step 6: Differentiate the right side using the product rule
For the right side, we apply the product rule:
\[ \frac{d}{dx} \left( (\tan x) \cdot \log(\tan x) \right) = \tan x \cdot \frac{d}{dx}(\log(\tan x)) + \log(\tan x) \cdot \frac{d}{dx}(\tan x) \]
Calculating \( \frac{d}{dx}(\log(\tan x)) \) and \( \frac{d}{dx}(\tan x) \):
\[ \frac{d}{dx}(\log(\tan x)) = \frac{1}{\tan x} \cdot \sec^2 x \]
\[ \frac{d}{dx}(\tan x) = \sec^2 x \]

### Step 7: Substitute back into the equation
Substituting these derivatives back gives:
\[ \frac{dy}{dx} = y \cdot \log y \left( \tan x \cdot \left( \frac{1}{\tan x} \sec^2 x \right) + \log(\tan x) \cdot \sec^2 x \right) \]
This simplifies to:
\[ \frac{dy}{dx} = y \cdot \log y \cdot \sec^2 x \left( 1 + \log(\tan x) \right) \]

### Step 8: Final expression
Thus, the final expression for \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = y \cdot \log y \cdot \sec^2 x \cdot \left( 1 + \log(\tan x) \right) \]