`d/(dx)(sin^(n)x)=…………………. .`

Answer

To differentiate \( \sin^n x \) with respect to \( x \), we will use the chain rule and the power rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
We have the function \( y = \sin^n x \).

### Step 2: Apply the chain rule
According to the chain rule, if we have a function \( f(g(x)) \), then the derivative is given by:
\[ \frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x) \]
In our case, let \( f(u) = u^n \) where \( u = \sin x \). Thus, we can express the derivative as:
\[ \frac{d}{dx} (\sin^n x) = \frac{d}{du} (u^n) \cdot \frac{d}{dx} (\sin x) \]

### Step 3: Differentiate \( f(u) = u^n \)
Using the power rule, we find:
\[ \frac{d}{du} (u^n) = n u^{n-1} \]
Substituting \( u = \sin x \):
\[ \frac{d}{du} (\sin^n x) = n \sin^{n-1} x \]

### Step 4: Differentiate \( \sin x \)
The derivative of \( \sin x \) is:
\[ \frac{d}{dx} (\sin x) = \cos x \]

### Step 5: Combine the results
Now, substituting back into our expression from Step 2:
\[ \frac{d}{dx} (\sin^n x) = n \sin^{n-1} x \cdot \cos x \]

### Final Result
Thus, the derivative of \( \sin^n x \) with respect to \( x \) is:
\[ \frac{d}{dx} (\sin^n x) = n \sin^{n-1} x \cos x \]

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