`sin^4x`

Answer

To find the derivative of the function \( f(x) = \sin^4 x \), we will use the chain rule. Here’s a step-by-step solution:

### Step 1: Identify the outer and inner functions
The function can be expressed as:
\[ f(x) = (\sin x)^4 \]
Here, the outer function is \( u^4 \) where \( u = \sin x \), and the inner function is \( \sin x \).

### Step 2: Differentiate the outer function
Using the power rule, the derivative of \( u^n \) is \( n \cdot u^{n-1} \). Therefore, the derivative of \( u^4 \) is:
\[ \frac{d}{du}(u^4) = 4u^3 \]

### Step 3: Differentiate the inner function
The derivative of \( \sin x \) is:
\[ \frac{d}{dx}(\sin x) = \cos x \]

### Step 4: Apply the chain rule
According to the chain rule:
\[ f'(x) = \frac{d}{du}(u^4) \cdot \frac{d}{dx}(\sin x) \]
Substituting the derivatives we found:
\[ f'(x) = 4(\sin x)^3 \cdot \cos x \]

### Step 5: Write the final answer
Thus, the derivative of \( f(x) = \sin^4 x \) is:
\[ f'(x) = 4 \sin^3 x \cos x \]

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