Find the derivative of `(x-1)(x+1)(x^(2)+1)(x^(4)+1)` with respect to x.

Answer

To find the derivative of the expression \( (x-1)(x+1)(x^2+1)(x^4+1) \) with respect to \( x \), we can follow these steps:

### Step 1: Simplify the Expression
Let \( y = (x-1)(x+1)(x^2+1)(x^4+1) \).

First, we can simplify \( (x-1)(x+1) \):
\[ (x-1)(x+1) = x^2 - 1 \]
Thus, we can rewrite \( y \) as:
\[ y = (x^2 - 1)(x^2 + 1)(x^4 + 1) \]

### Step 2: Further Simplify
Next, we simplify \( (x^2 - 1)(x^2 + 1) \):
\[ (x^2 - 1)(x^2 + 1) = x^4 - 1 \]
Now, we can rewrite \( y \) as:
\[ y = (x^4 - 1)(x^4 + 1) \]

### Step 3: Use Difference of Squares
Now, we can use the difference of squares on \( (x^4 - 1)(x^4 + 1) \):
\[ y = x^8 - 1 \]

### Step 4: Differentiate the Expression
Now that we have simplified \( y \) to \( y = x^8 - 1 \), we can differentiate it with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^8 - 1) \]

Using the power rule of differentiation:
\[ \frac{dy}{dx} = 8x^{8-1} - 0 = 8x^7 \]

### Final Answer
Thus, the derivative of the expression \( (x-1)(x+1)(x^2+1)(x^4+1) \) with respect to \( x \) is:
\[ \frac{dy}{dx} = 8x^7 \]

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