Evaluate differentiation of y with respect to x:
`sin^(5)x`

Explanation

To evaluate the differentiation of \( y \) with respect to \( x \) for the function \( y = \sin^5(x) \), we will use the chain rule of differentiation. Here’s a step-by-step solution:

### Step 1: Identify the function
We have:
\[ y = \sin^5(x) \]
This means that \( y \) is the sine function raised to the power of 5.

### Step 2: Apply the chain rule
To differentiate \( y \), we will use the chain rule. The chain rule states that if you have a composite function \( f(g(x)) \), then the derivative is given by:
\[ \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \]
In our case, let \( u = \sin(x) \), then \( y = u^5 \).

### Step 3: Differentiate the outer function
Now, differentiate the outer function \( y = u^5 \):
\[ \frac{dy}{du} = 5u^4 \]

### Step 4: Differentiate the inner function
Next, differentiate the inner function \( u = \sin(x) \):
\[ \frac{du}{dx} = \cos(x) \]

### Step 5: Combine the derivatives
Now, we will combine the derivatives using the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = 5u^4 \cdot \cos(x) \]
Substituting back \( u = \sin(x) \):
\[ \frac{dy}{dx} = 5\sin^4(x) \cdot \cos(x) \]

### Final Answer
Thus, the derivative of \( y = \sin^5(x) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 5\sin^4(x) \cos(x) \]

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