Differentiate w.r.t. x the function

x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

Let y = x^x + x^a + a^x + a^a, for some fixed a > 0 and x > 0

And let x^x = u, x^a = v, a^x = w and a^a = s

Then y = u + v + w + s

∴ ……..(I)

Now,

u = x^x

Taking logarithm both sides, we get

log u = log x^x

⇒ log u = x log x

Differentiating both sides w.r.t. x

⇒

⇒ …….(II)

v = x^a

Differentiating both sides with respect to x

⇒ ………..(III)

w = a^x

Taking logarithm both sides

log w = log a^x

log w = x log a

Differentiating both sides with respect to x

⇒

⇒ …………….(IV)

s = a^a

Differentiating both sides with respect to x

… ……….(V)

Putting (II), (III), (IV) and (V) in (I)

∴