Differentiate `(x^(8)-1)/(x^(4)-1)` with respect to x.

Answer

To differentiate the function \( f(x) = \frac{x^8 - 1}{x^4 - 1} \) with respect to \( x \), we can follow these steps:

### Step 1: Simplify the Function
First, we can simplify the function before differentiating it. Notice that \( x^8 - 1 \) can be factored using the difference of squares:

\[ x^8 - 1 = (x^4 - 1)(x^4 + 1) \]

Thus, we can rewrite \( f(x) \):

\[ f(x) = \frac{(x^4 - 1)(x^4 + 1)}{x^4 - 1} \]

### Step 2: Cancel Common Terms
Since \( x^4 - 1 \) appears in both the numerator and the denominator, we can cancel these terms (provided \( x^4 - 1 \neq 0 \)):

\[ f(x) = x^4 + 1 \]

### Step 3: Differentiate the Simplified Function
Now we differentiate \( f(x) = x^4 + 1 \):

Using the power rule of differentiation, which states that if \( f(x) = x^n \), then \( f'(x) = nx^{n-1} \):

\[ f'(x) = \frac{d}{dx}(x^4) + \frac{d}{dx}(1) = 4x^{4-1} + 0 = 4x^3 \]

### Final Answer
Thus, the derivative of the function \( f(x) \) with respect to \( x \) is:

\[ f'(x) = 4x^3 \]

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