If ` y= (tan x )^(sin x ) ,then (dy)/(dx)=`

Options

A. ` y( sec ^(2)x +cos x )log (tan x )) `
B. ` y(sec ^(2) x-(cosx) log (tan x) ) `
C. ` y(sec x+(cosx) log (tan x) ) `
D. ` y(sec x-(cosx) log (tan x) ) `

Answer

To find the derivative of \( y = (\tan x)^{\sin x} \), we will use logarithmic differentiation. Here’s a step-by-step solution:

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

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

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

\[ \ln y = \sin x \cdot \ln(\tan x) \]

### Step 3: Differentiate both sides
Now we differentiate both sides with respect to \( x \). We will use the product rule on the right side, where \( u = \sin x \) and \( v = \ln(\tan x) \).

The derivative of \( \ln y \) with respect to \( x \) is:

\[ \frac{1}{y} \frac{dy}{dx} \]

For the right-hand side, using the product rule:

\[ \frac{d}{dx}(\sin x \cdot \ln(\tan x)) = \cos x \cdot \ln(\tan x) + \sin x \cdot \frac{d}{dx}(\ln(\tan x)) \]

### Step 4: Differentiate \( \ln(\tan x) \)
Now we need to differentiate \( \ln(\tan x) \):

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

Thus, we have:

\[ \frac{d}{dx}(\ln(\tan x)) = \frac{\sec^2 x}{\tan x} = \frac{1}{\sin x \cos x} = \csc x \sec x \]

### Step 5: Substitute back into the derivative
Now we can substitute this back into our differentiation result:

\[ \frac{1}{y} \frac{dy}{dx} = \cos x \cdot \ln(\tan x) + \sin x \cdot \csc x \sec x \]

### Step 6: Simplify the equation
Since \( \sin x \cdot \csc x = 1 \), we can simplify the equation:

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

### Step 7: Solve for \( \frac{dy}{dx} \)
Now, multiply both sides by \( y \):

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

### Step 8: Substitute back for \( y \)
Recall that \( y = (\tan x)^{\sin x} \):

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

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

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

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