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

Options

A. ` x^(sinx ) ((cosx )( log x) -(sin x)/( x )) + (sin x)^(x) (log sin x -xcot x ) `
B. ` x^(sinx ) ((cosx )( log x) +(sin x)/( x )) + (sin x)^(x) (log sin x +xcot x ) `
C. ` x^(sinx ) ((cosx )( log x) -(sin x)/( x )) - (sin x)^(x) (log sin x +xcot x ) `
D. ` x^(sinx ) ((cosx )( log x) +(sin x)/( x )) + (sin x)^(x) (log sin x -xcot x ) `

Answer

To find the derivative \( \frac{dy}{dx} \) for the function \( y = x^{\sin x} + (\sin x)^x \), we will differentiate each part of the function separately.

### Step-by-Step Solution

1. **Identify the function**:
\[ y = x^{\sin x} + (\sin x)^x \]

2. **Differentiate the first term \( x^{\sin x} \)**:
- Let \( u = x^{\sin x} \).
- Take the natural logarithm of both sides:
\[ \ln u = \sin x \cdot \ln x \]
- Differentiate both sides using implicit differentiation:
\[ \frac{1}{u} \frac{du}{dx} = \cos x \cdot \ln x + \sin x \cdot \frac{1}{x} \]
- Multiply both sides by \( u \):
\[ \frac{du}{dx} = u \left( \cos x \cdot \ln x + \frac{\sin x}{x} \right) \]
- Substitute \( u = x^{\sin x} \):
\[ \frac{du}{dx} = x^{\sin x} \left( \cos x \cdot \ln x + \frac{\sin x}{x} \right) \]

3. **Differentiate the second term \( (\sin x)^x \)**:
- Let \( v = (\sin x)^x \).
- Take the natural logarithm of both sides:
\[ \ln v = x \cdot \ln(\sin x) \]
- Differentiate both sides:
\[ \frac{1}{v} \frac{dv}{dx} = \ln(\sin x) + x \cdot \frac{\cos x}{\sin x} \]
- Multiply both sides by \( v \):
\[ \frac{dv}{dx} = v \left( \ln(\sin x) + x \cdot \cot x \right) \]
- Substitute \( v = (\sin x)^x \):
\[ \frac{dv}{dx} = (\sin x)^x \left( \ln(\sin x) + x \cdot \cot x \right) \]

4. **Combine the derivatives**:
- Now, we can find \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{du}{dx} + \frac{dv}{dx} \]
- Substitute the expressions for \( \frac{du}{dx} \) and \( \frac{dv}{dx} \):
\[ \frac{dy}{dx} = x^{\sin x} \left( \cos x \cdot \ln x + \frac{\sin x}{x} \right) + (\sin x)^x \left( \ln(\sin x) + x \cdot \cot x \right) \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is given by:
\[ \frac{dy}{dx} = x^{\sin x} \left( \cos x \cdot \ln x + \frac{\sin x}{x} \right) + (\sin x)^x \left( \ln(\sin x) + x \cdot \cot x \right) \]