If `y = (1)/(1 + x^(a - b) + x^(c - b)) + (1)/(1 + x^(b-c) + x^(a - c)) + (1)/(1 + x^(b - a) + x^(c - a))` then find `(dy)/(dx)` at `e^(a^(b^(c )))`

Answer

To solve the problem, we need to find the derivative of the function \( y \) given by:

\[ y = \frac{1}{1 + x^{a - b} + x^{c - b}} + \frac{1}{1 + x^{b - c} + x^{a - c}} + \frac{1}{1 + x^{b - a} + x^{c - a}} \]

and then evaluate \( \frac{dy}{dx} \) at \( x = e^{a^{b^{c}}} \).

### Step 1: Simplifying the Expression for \( y \)

We can rewrite each term in \( y \):

1. The first term:
\[ \frac{1}{1 + x^{a - b} + x^{c - b}} \]
2. The second term:
\[ \frac{1}{1 + x^{b - c} + x^{a - c}} \]
3. The third term:
\[ \frac{1}{1 + x^{b - a} + x^{c - a}} \]

### Step 2: Finding a Common Denominator

The common denominator for all three fractions is:
\[ (1 + x^{a - b} + x^{c - b})(1 + x^{b - c} + x^{a - c})(1 + x^{b - a} + x^{c - a}) \]

However, we can observe that the structure of the terms suggests that they might simplify significantly.

### Step 3: Observing the Pattern

If we add the three fractions, we notice that each term has a similar structure. The sum can be simplified as follows:

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

### Step 4: Simplifying Further

After simplification, we find that the numerator and denominator yield the same expression, leading to:

\[ y = 1 \]

### Step 5: Finding the Derivative \( \frac{dy}{dx} \)

Since \( y \) is a constant (1), the derivative with respect to \( x \) is:

\[ \frac{dy}{dx} = 0 \]

### Step 6: Evaluating at \( x = e^{a^{b^{c}}} \)

Since \( \frac{dy}{dx} = 0 \), it holds true for all values of \( x \), including \( x = e^{a^{b^{c}}} \).

### Final Answer

Thus, we conclude that:

\[ \frac{dy}{dx} \text{ at } x = e^{a^{b^{c}}} = 0 \]

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