If `f(x)` is a function satisfying `f(x)-f(y)=(y^y)/(x^x)f((x^x)/(y^y))AAx , y in R^*"and"f^(prime)(1)=1` , then find `f(x)`

Answer

To solve the problem, we need to find the function \( f(x) \) given the condition:

\[ f(x) - f(y) = \frac{y^y}{x^x} f\left(\frac{x^x}{y^y}\right) \]

and the derivative condition \( f'(1) = 1 \).

### Step 1: Analyze the given functional equation

We start with the functional equation:

\[ f(x) - f(y) = \frac{y^y}{x^x} f\left(\frac{x^x}{y^y}\right) \]

This suggests a relationship between the values of the function at different points. A common approach is to consider specific values for \( x \) and \( y \).

### Step 2: Substitute specific values

Let’s set \( y = 1 \):

\[ f(x) - f(1) = \frac{1^1}{x^x} f\left(\frac{x^x}{1^1}\right) \]

This simplifies to:

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

### Step 3: Rearranging the equation

Rearranging gives us:

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

### Step 4: Guessing a form for \( f(x) \)

We can guess that \( f(x) \) might have the form \( f(x) = a \ln x \) for some constant \( a \). This is a common form for functions that involve logarithmic properties.

### Step 5: Substitute \( f(x) = a \ln x \)

Substituting \( f(x) = a \ln x \) into the equation:

\[ f(x^x) = a \ln(x^x) = a \cdot x \ln x \]

Now substituting back into our rearranged equation:

\[ f(x) = f(1) + \frac{1}{x^x} (a \cdot x \ln x) \]

### Step 6: Calculate \( f(1) \)

We know \( f(1) = a \ln 1 = 0 \). Thus, we have:

\[ f(x) = \frac{a \cdot x \ln x}{x^x} \]

### Step 7: Simplifying further

Since \( x^x = e^{x \ln x} \), we can express:

\[ f(x) = a \cdot x \ln x \cdot e^{-x \ln x} = a \cdot \frac{\ln x}{x^{x-1}} \]

### Step 8: Find \( a \) using the derivative condition

We need to satisfy \( f'(1) = 1 \). First, we differentiate \( f(x) \):

\[ f'(x) = a \left( \frac{1}{x} + \ln x \cdot \frac{d}{dx}(x^{x-1}) \right) \]

At \( x = 1 \):

\[ f'(1) = a \left( 1 + 0 \right) = a \]

Setting \( f'(1) = 1 \) gives us \( a = 1 \).

### Step 9: Final function

Thus, we find:

\[ f(x) = \ln x \]

### Conclusion

The function \( f(x) \) that satisfies the given conditions is:

\[ \boxed{f(x) = \ln x} \]