If y^x + x^y + x^x = a^b, find

To find:

Taking given equation y^x + x^y + x^x = a^b

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

∴ Our given equation becomes u + v + w = a^b

…(A)

Now, u = y^x

Taking log on both the sides, we get

log u = x logy

Differentiating with respect to x, we get

…(i) [replacing u by y^x]

Now, v = x^y

Taking log on both the sides, we get

log v = y logx

Differentiating with respect to x, we get

…(ii) [replacing v by x^y]

Also, w = x^x

Taking log on both the sides, we get

log w = x logx

Differentiating with respect to x, we get

…(iii) [replacing w by x^x]

Substituting the value of eq. (i), (ii) and (iii) in eq. (A), we get