If x^y + y^x = a, where ‘a’ is a constant. Find .

Given: x^y + y^x = a where ‘a’ is a constant

To find:

x^y + y^x = a

{∵ log e = 1}

Differentiating both sides with respect to x:

Let u = x^y and v = y^x

u = x^y

Taking log both sides:

⇒ log u = log (x^y)

⇒ log u = y log x

{∵ log a^x = x log a}

Differentiating both sides with respect to x:

Put u = x^y:

v = y^x

Taking log both sides:

⇒ log v = log (y^x)

⇒ log v = x log y

{∵ log a^x = x log a}

Differentiating both sides with respect to x:

Put v = y^x:

Now,

From (1) and (2):