Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers) :
`sin^(n)x`

Answer

To find the derivative of the function \( y = \sin^n x \), we will use the chain rule of differentiation. Here are the steps:

### Step-by-Step Solution:

1. **Define the function**:
Let \( y = \sin^n x \).

2. **Differentiate using the chain rule**:
To differentiate \( y \) with respect to \( x \), we apply the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is \( f'(g(x)) \cdot g'(x) \).

Here, we can consider \( f(u) = u^n \) where \( u = \sin x \). Thus, we have:
\[ \frac{dy}{dx} = \frac{d}{dx}(\sin^n x) = n \sin^{n-1} x \cdot \frac{d}{dx}(\sin x) \]

3. **Differentiate \( \sin x \)**:
The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \). Therefore, substituting this into our equation gives:
\[ \frac{dy}{dx} = n \sin^{n-1} x \cdot \cos x \]

4. **Final result**:
Thus, the derivative of \( y = \sin^n x \) is:
\[ \frac{dy}{dx} = n \sin^{n-1} x \cos x \]