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Question
Differentiate the function with respect to x:
xx + xa + ax + aa, for some fixed a > 0 and x > 0
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Solution
Let, y = xx + xa + ax + aa
On differentiating with respect to x,
`dy/dx = d/dx (x^x) + d/dx (x^a) + d/dx (a^x) + (a^a) d/dx (1)`
= `d/dx (x^x) + ax^(a - 1) + a^x log a + 0` ...(1)
Let, u = xx
Taking log on both sides,
log u = x log x
On differentiating with respect to x,
`1/u (du)/dx = x d/dx log x + log x d/dx (x)`
= `x * 1/x + log x`
= (1 + log x)
∴ `(du)/dx` = u (1 + log x)
= xx (1 + log x)
i.e., `d/dx (x^x) = (du)/dx`
= xx (1 + log x)
Putting the value of `d/dx (x^x)` in equation (1),
`dy/dx` = xx (1 + log x) + axa − 1 + ax log a
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