Differentiate the following w.r.t. x :
`x^(sin2x+cos2x)`

Answer

To differentiate the function \( y = x^{\sin(2x) + \cos(2x)} \) 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:
\[ \ln(y) = \ln(x^{\sin(2x) + \cos(2x)}) \]

### Step 2: Simplify using properties of logarithms
Using the property of logarithms that states \( \ln(a^b) = b \ln(a) \), we can simplify the right-hand side:
\[ \ln(y) = (\sin(2x) + \cos(2x)) \ln(x) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \). We will use the product rule on the right-hand side:
\[ \frac{d}{dx}(\ln(y)) = \frac{1}{y} \frac{dy}{dx} \]
For the right-hand side, we apply the product rule:
\[ \frac{d}{dx}((\sin(2x) + \cos(2x)) \ln(x)) = \frac{d}{dx}(\sin(2x) + \cos(2x)) \cdot \ln(x) + (\sin(2x) + \cos(2x)) \cdot \frac{d}{dx}(\ln(x)) \]

### Step 4: Differentiate \( \sin(2x) + \cos(2x) \)
The derivative of \( \sin(2x) + \cos(2x) \) is:
\[ \frac{d}{dx}(\sin(2x)) = 2\cos(2x) \quad \text{and} \quad \frac{d}{dx}(\cos(2x)) = -2\sin(2x) \]
Thus,
\[ \frac{d}{dx}(\sin(2x) + \cos(2x)) = 2\cos(2x) - 2\sin(2x) \]

### Step 5: Differentiate \( \ln(x) \)
The derivative of \( \ln(x) \) is:
\[ \frac{d}{dx}(\ln(x)) = \frac{1}{x} \]

### Step 6: Combine the derivatives
Now we can combine everything:
\[ \frac{1}{y} \frac{dy}{dx} = (2\cos(2x) - 2\sin(2x)) \ln(x) + (\sin(2x) + \cos(2x)) \cdot \frac{1}{x} \]

### Step 7: Solve for \( \frac{dy}{dx} \)
Multiplying both sides by \( y \) gives:
\[ \frac{dy}{dx} = y \left( (2\cos(2x) - 2\sin(2x)) \ln(x) + \frac{\sin(2x) + \cos(2x)}{x} \right) \]

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

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