Differentiate `(1)/(1+x^(b-a)+x^(c-a))+(1)/(1+x^(a-b)+x^(c-b))+(1)/(1+x^(a-c)+x^(b-c))` w.r.t. to x.

Answer

To differentiate the given expression

\[ f(x) = \frac{1}{1+x^{b-a}+x^{c-a}} + \frac{1}{1+x^{a-b}+x^{c-b}} + \frac{1}{1+x^{a-c}+x^{b-c}} \]

with respect to \(x\), we will simplify the expression first before differentiating.

### Step 1: Simplify the Expression

We can rewrite each term in the expression:

1. The first term can be rewritten as:
\[ \frac{1}{1+x^{b-a}+x^{c-a}} = \frac{1}{1 + \frac{x^{b}}{x^{a}} + \frac{x^{c}}{x^{a}}} = \frac{x^{a}}{x^{a} + x^{b} + x^{c}} \]

2. The second term can be rewritten as:
\[ \frac{1}{1+x^{a-b}+x^{c-b}} = \frac{1}{1 + \frac{x^{a}}{x^{b}} + \frac{x^{c}}{x^{b}}} = \frac{x^{b}}{x^{a} + x^{b} + x^{c}} \]

3. The third term can be rewritten as:
\[ \frac{1}{1+x^{a-c}+x^{b-c}} = \frac{1}{1 + \frac{x^{a}}{x^{c}} + \frac{x^{b}}{x^{c}}} = \frac{x^{c}}{x^{a} + x^{b} + x^{c}} \]

Now, combining these three terms, we have:

\[ f(x) = \frac{x^{a}}{x^{a} + x^{b} + x^{c}} + \frac{x^{b}}{x^{a} + x^{b} + x^{c}} + \frac{x^{c}}{x^{a} + x^{b} + x^{c}} \]

### Step 2: Combine the Terms

Since all three fractions have the same denominator, we can combine them:

\[ f(x) = \frac{x^{a} + x^{b} + x^{c}}{x^{a} + x^{b} + x^{c}} = 1 \]

### Step 3: Differentiate the Simplified Expression

Now, we differentiate \(f(x)\):

\[ f'(x) = \frac{d}{dx}(1) = 0 \]

### Final Answer

Thus, the derivative of the given expression with respect to \(x\) is:

\[ \boxed{0} \]