Evaluate differentiation of y with respect to x :
`y=sinxcosx`

Explanation

To evaluate the differentiation of \( y \) with respect to \( x \) where \( y = \sin x \cos x \), we will follow these steps:

### Step 1: Identify the function
We have:
\[ y = \sin x \cos x \]

### Step 2: Apply the product rule
Since \( y \) is a product of two functions \( \sin x \) and \( \cos x \), we will use the product rule for differentiation. The product rule states that if \( u = \sin x \) and \( v = \cos x \), then:
\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

### Step 3: Differentiate each function
We need to find the derivatives of \( u \) and \( v \):
- The derivative of \( \sin x \) is \( \cos x \):
\[ \frac{du}{dx} = \cos x \]
- The derivative of \( \cos x \) is \( -\sin x \):
\[ \frac{dv}{dx} = -\sin x \]

### Step 4: Substitute into the product rule
Now substituting back into the product rule:
\[ \frac{dy}{dx} = \sin x \cdot (-\sin x) + \cos x \cdot \cos x \]

### Step 5: Simplify the expression
This simplifies to:
\[ \frac{dy}{dx} = -\sin^2 x + \cos^2 x \]

### Step 6: Rearranging the terms
We can rearrange the terms to express it more clearly:
\[ \frac{dy}{dx} = \cos^2 x - \sin^2 x \]

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

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