`y=x^(tanx)`

Answer

To differentiate the function \( y = x^{\tan x} \), 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 differentiation process.

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

### Step 2: Apply the logarithmic identity
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can rewrite the equation:

\[ \ln y = \tan x \cdot \ln x \]

### Step 3: Differentiate both sides
Now we differentiate both sides with respect to \( x \). We will use implicit differentiation on the left side and the product rule on the right side.

\[ \frac{d}{dx}(\ln y) = \frac{d}{dx}(\tan x \cdot \ln x) \]

Using the chain rule on the left side:

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

### Step 4: Differentiate the right side
Now we differentiate the right side. The derivative of \( \tan x \) is \( \sec^2 x \) and the derivative of \( \ln x \) is \( \frac{1}{x} \):

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

### Step 5: Solve for \( \frac{dy}{dx} \)
Now we multiply both sides by \( y \) to solve for \( \frac{dy}{dx} \):

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

### Step 6: Substitute \( y \) back
Since we have \( y = x^{\tan x} \), we substitute back:

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

### Final Result
Thus, the derivative of \( y = x^{\tan x} \) is:

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

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