Let `y` be an implicit function of `x` defined by `x^(2x)-2x^xcot y-1=0.` Then `y '(1)` equals: `1` b. `log2` c. `-log2` d. `-1`

Options

A. -1
B. 1
C. log 2
D. `-log 2

Explanation

To solve the problem, we need to find the derivative \( \frac{dy}{dx} \) at \( x = 1 \) for the implicit function defined by the equation:

\[ x^{2x} - 2x^x \cot y - 1 = 0 \]

### Step 1: Substitute \( x = 1 \) to find \( y \)

First, we substitute \( x = 1 \) into the equation:

\[ 1^{2 \cdot 1} - 2 \cdot 1^1 \cot y - 1 = 0 \]

This simplifies to:

\[ 1 - 2 \cot y - 1 = 0 \]

This further simplifies to:

\[ -2 \cot y = 0 \]

Thus, we have:

\[ \cot y = 0 \]

The solution to \( \cot y = 0 \) is:

\[ y = \frac{\pi}{2} \]

### Step 2: Differentiate the equation implicitly

Now, we differentiate the original equation with respect to \( x \):

\[ \frac{d}{dx}(x^{2x}) - \frac{d}{dx}(2x^x \cot y) - \frac{d}{dx}(1) = 0 \]

### Step 3: Differentiate \( x^{2x} \)

Using the product rule and chain rule, we differentiate \( x^{2x} \):

Let \( u = x^{2x} \). Then,

\[ \frac{du}{dx} = x^{2x} \left( 2 \ln x + 2 \right) \]

### Step 4: Differentiate \( 2x^x \cot y \)

Now we differentiate \( 2x^x \cot y \):

Using the product rule:

\[ \frac{d}{dx}(2x^x \cot y) = 2 \left( x^x \ln x \cot y + x^x \frac{d(\cot y)}{dx} \right) \]

Using the chain rule for \( \cot y \):

\[ \frac{d(\cot y)}{dx} = -\csc^2 y \frac{dy}{dx} \]

Thus, we have:

\[ \frac{d}{dx}(2x^x \cot y) = 2 \left( x^x \ln x \cot y - x^x \csc^2 y \frac{dy}{dx} \right) \]

### Step 5: Substitute back into the differentiated equation

Now substituting back into the differentiated equation:

\[ x^{2x} (2 \ln x + 2) - 2 \left( x^x \ln x \cot y - x^x \csc^2 y \frac{dy}{dx} \right) = 0 \]

### Step 6: Evaluate at \( x = 1 \) and \( y = \frac{\pi}{2} \)

Substituting \( x = 1 \) and \( y = \frac{\pi}{2} \):

1. \( x^{2x} = 1^{2 \cdot 1} = 1 \)
2. \( \ln 1 = 0 \)
3. \( \cot \frac{\pi}{2} = 0 \)
4. \( \csc^2 \frac{\pi}{2} = 1 \)

Thus, the equation simplifies to:

\[ 1(2 \cdot 0 + 2) - 2(1 \cdot 0 - 1 \cdot \frac{dy}{dx}) = 0 \]

This simplifies to:

\[ 2 + 2 \frac{dy}{dx} = 0 \]

### Step 7: Solve for \( \frac{dy}{dx} \)

Rearranging gives:

\[ 2 \frac{dy}{dx} = -2 \]

Thus,

\[ \frac{dy}{dx} = -1 \]

### Final Answer

So, the value of \( \frac{dy}{dx} \) at \( x = 1 \) is:

\[ \frac{dy}{dx} = -1 \]

### Conclusion

The correct option is:

**d. -1**