Given equation:
x^y * y^x = 25

To find dy/dx at (1,1):
To find the derivative of y with respect to x, we can use implicit differentiation. Let's differentiate both sides of the equation with respect to x.

Step 1: Take the natural logarithm of both sides
Taking the natural logarithm helps simplify the equation and make it easier to differentiate. Applying the logarithm rules, we have:
ln(x^y * y^x) = ln(25)

Step 2: Apply logarithm rules
Using the logarithm rules, we can simplify the left side of the equation:
ln(x^y) + ln(y^x) = ln(25)

Step 3: Differentiate both sides
Now, we differentiate both sides of the equation with respect to x using the chain rule and the logarithmic differentiation rule.

Differentiating the left side:
d/dx [ln(x^y) + ln(y^x)] = d/dx [ln(25)]

Applying the logarithmic differentiation rule:
(1/x^y) * y + (1/y^x) * x * dy/dx = 0

Step 4: Substitute the values of x and y
We are given the point (1,1) at which we need to find dy/dx. Substituting x = 1 and y = 1 into the equation, we have:
(1/1^1) * 1 + (1/1^1) * 1 * dy/dx = 0

Simplifying the equation:
1 + 1 * dy/dx = 0
1 + dy/dx = 0

Step 5: Solve for dy/dx
To find dy/dx, we isolate the variable:
dy/dx = -1

Conclusion:
The derivative dy/dx at the point (1,1) is -1.