Differentiate the following w.r.t.
`(x-(1)/(x))(x+(1)/(x))(x^(2)+(1)/(x^(2)))(x^(4)+(1)/(x^(4)))`

Answer

To differentiate the expression \( y = (x - \frac{1}{x})(x + \frac{1}{x})(x^2 + \frac{1}{x^2})(x^4 + \frac{1}{x^4}) \) with respect to \( x \), we will first simplify the expression using algebraic identities before differentiating.

### Step 1: Simplify the expression

We can simplify the expression using the identity \( (a - b)(a + b) = a^2 - b^2 \).

1. Let \( a = x \) and \( b = \frac{1}{x} \).
2. The first part:
\[ (x - \frac{1}{x})(x + \frac{1}{x}) = x^2 - \left(\frac{1}{x}\right)^2 = x^2 - \frac{1}{x^2} \]

3. Now, we have:
\[ y = (x^2 - \frac{1}{x^2})(x^2 + \frac{1}{x^2})(x^4 + \frac{1}{x^4}) \]

4. Next, we simplify \( (x^2 - \frac{1}{x^2})(x^2 + \frac{1}{x^2}) \):
\[ (x^2 - \frac{1}{x^2})(x^2 + \frac{1}{x^2}) = (x^2)^2 - \left(\frac{1}{x^2}\right)^2 = x^4 - \frac{1}{x^4} \]

5. Now we have:
\[ y = (x^4 - \frac{1}{x^4})(x^4 + \frac{1}{x^4}) \]

6. Finally, we simplify \( (x^4 - \frac{1}{x^4})(x^4 + \frac{1}{x^4}) \):
\[ (x^4 - \frac{1}{x^4})(x^4 + \frac{1}{x^4}) = (x^4)^2 - \left(\frac{1}{x^4}\right)^2 = x^8 - \frac{1}{x^8} \]

Thus, we have simplified the expression to:
\[ y = x^8 - \frac{1}{x^8} \]

### Step 2: Differentiate the expression

Now we differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{d}{dx}(x^8) - \frac{d}{dx}\left(\frac{1}{x^8}\right) \]

1. For the first term, using the power rule:
\[ \frac{d}{dx}(x^8) = 8x^{7} \]

2. For the second term, we rewrite it as \( x^{-8} \):
\[ \frac{d}{dx}\left(x^{-8}\right) = -8x^{-9} \]

So, combining these results:
\[ \frac{dy}{dx} = 8x^{7} + 8x^{-9} \]

### Final Answer
Thus, the derivative of the given expression is:
\[ \frac{dy}{dx} = 8x^{7} + \frac{8}{x^{9}} \]