Differentiate the following w.r.t. x :
`(sinx)^(tanx)`

Answer

To differentiate the function \( y = (\sin x)^{\tan x} \) with respect to \( x \), we will use logarithmic differentiation. Here are the 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:
\[ \ln y = \ln((\sin x)^{\tan x}) \]

### Step 2: Apply the logarithmic identity
Using the property of logarithms that states \( \ln(a^b) = b \ln a \), we can rewrite the equation:
\[ \ln y = \tan x \cdot \ln(\sin x) \]

### Step 3: Differentiate both sides
Now, we differentiate both sides with respect to \( x \). For the left side, we apply the chain rule:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{d}{dx}(\tan x \cdot \ln(\sin x)) \]

### Step 4: Use the product rule on the right side
Now we need to differentiate the right side using the product rule:
\[ \frac{d}{dx}(\tan x \cdot \ln(\sin x)) = \tan x \cdot \frac{d}{dx}(\ln(\sin x)) + \ln(\sin x) \cdot \frac{d}{dx}(\tan x) \]

### Step 5: Differentiate \( \ln(\sin x) \) and \( \tan x \)
Now we calculate the derivatives:
- The derivative of \( \ln(\sin x) \) is \( \frac{1}{\sin x} \cdot \cos x = \cot x \).
- The derivative of \( \tan x \) is \( \sec^2 x \).

Substituting these into our equation gives:
\[ \frac{1}{y} \frac{dy}{dx} = \tan x \cdot \cot x + \ln(\sin x) \cdot \sec^2 x \]

### Step 6: Simplify the equation
Since \( \tan x \cdot \cot x = 1 \), we can simplify:
\[ \frac{1}{y} \frac{dy}{dx} = 1 + \ln(\sin x) \cdot \sec^2 x \]

### Step 7: Solve for \( \frac{dy}{dx} \)
Now, multiply both sides by \( y \) to isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = y \left( 1 + \ln(\sin x) \cdot \sec^2 x \right) \]

### Step 8: Substitute back for \( y \)
Recall that \( y = (\sin x)^{\tan x} \):
\[ \frac{dy}{dx} = (\sin x)^{\tan x} \left( 1 + \ln(\sin x) \cdot \sec^2 x \right) \]

### Final Answer
Thus, the derivative of \( y = (\sin x)^{\tan x} \) with respect to \( x \) is:
\[ \frac{dy}{dx} = (\sin x)^{\tan x} \left( 1 + \ln(\sin x) \cdot \sec^2 x \right) \]