If `y = sin (x^(x)) " then " (dy)/(dx) =` ?

Options

A. `x^(x) cos (x^(x))`
B. `x^(x) cos x^(x) (1 + log x)`
C. `x^(x) cos x^(x) log x`
D. none of these

Explanation

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

### Step 1: Differentiate \( y = \sin(x^x) \)

Using the chain rule, we differentiate \( y \) with respect to \( x \):

\[ \frac{dy}{dx} = \cos(x^x) \cdot \frac{d}{dx}(x^x) \]

### Step 2: Differentiate \( x^x \)

To differentiate \( x^x \), we will use logarithmic differentiation. Let \( p = x^x \). Taking the natural logarithm of both sides gives:

\[ \ln(p) = x \ln(x) \]

### Step 3: Differentiate both sides

Now, we differentiate both sides with respect to \( x \):

\[ \frac{1}{p} \frac{dp}{dx} = \ln(x) + 1 \]

### Step 4: Solve for \( \frac{dp}{dx} \)

Multiply both sides by \( p \):

\[ \frac{dp}{dx} = p(\ln(x) + 1) \]

Substituting back \( p = x^x \):

\[ \frac{dp}{dx} = x^x (\ln(x) + 1) \]

### Step 5: Substitute back into \( \frac{dy}{dx} \)

Now substitute \( \frac{dp}{dx} \) back into the derivative of \( y \):

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

### Final Answer

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

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

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