Differentiate `sin^(m)x*cos^(n)x`

Explanation

To differentiate the function \( y = \sin^m x \cdot \cos^n x \), we will use the product rule and the chain rule. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Identify the function**:
\[ y = \sin^m x \cdot \cos^n x \]

2. **Apply the product rule**:
The product rule states that if you have two functions \( u \) and \( v \), then the derivative \( \frac{d}{dx}(uv) = u'v + uv' \). Here, let:
\[ u = \sin^m x \quad \text{and} \quad v = \cos^n x \]
Therefore, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = u'v + uv' \]

3. **Differentiate \( u \) and \( v \)**:
- For \( u = \sin^m x \), we use the chain rule:
\[ u' = m \sin^{m-1} x \cdot \cos x \]
- For \( v = \cos^n x \), we also use the chain rule:
\[ v' = n \cos^{n-1} x \cdot (-\sin x) = -n \cos^{n-1} x \cdot \sin x \]

4. **Substitute \( u' \) and \( v' \) back into the product rule**:
\[ \frac{dy}{dx} = (m \sin^{m-1} x \cdot \cos x) \cdot \cos^n x + \sin^m x \cdot (-n \cos^{n-1} x \cdot \sin x) \]

5. **Simplify the expression**:
\[ \frac{dy}{dx} = m \sin^{m-1} x \cdot \cos^{n+1} x - n \sin^{m+1} x \cdot \cos^{n-1} x \]

6. **Final expression**:
\[ \frac{dy}{dx} = m \sin^{m-1} x \cdot \cos^{n+1} x - n \sin^{m+1} x \cdot \cos^{n-1} x \]