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 can follow these steps:

### Step 1: Rewrite the function
Let \( y = \sin^5 x \).

### Step 2: Use the chain rule
To differentiate \( y \) with respect to \( x \), we can use the chain rule. We can express \( y \) in terms of a new variable \( t \):
Let \( t = \sin x \). Then, we can rewrite \( y \) as:
\[ y = t^5 \]

### Step 3: Differentiate \( y \) with respect to \( t \)
Now, we differentiate \( y \) with respect to \( t \):
\[ \frac{dy}{dt} = 5t^4 \]

### Step 4: Differentiate \( t \) with respect to \( x \)
Next, we need to differentiate \( t \) with respect to \( x \):
\[ \frac{dt}{dx} = \cos x \]

### Step 5: Apply the chain rule
Now, we can apply the chain rule:
\[ \frac{dy}{dx} = \frac{dy}{dt} \cdot \frac{dt}{dx} \]
Substituting the derivatives we found:
\[ \frac{dy}{dx} = 5t^4 \cdot \cos x \]

### Step 6: Substitute back for \( t \)
Now, substitute back \( t = \sin x \):
\[ \frac{dy}{dx} = 5(\sin x)^4 \cdot \cos x \]

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

---