If y = (xx)x, then what is the value of  at x = 1?

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NDA 01/2022: Maths Previous Year paper (Held On 10 April 2022)
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Answer (Detailed Solution Below)

Option 2 : 1
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NDA 01/2025: English Subject Test
30 Qs. 120 Marks 30 Mins

Detailed Solution

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Formula used : 

  1. log mn = n log m
  2. Differentiation by part: 

 

Calculation :

Given that y = (xx)x,

Taking log on both sides

log y = log (xx)x

Using the formula (1)

⇒ log y = x log xx

⇒ log y = x2 log x

Diff. with respect to x 

(log y) = (x2log x)

Doing diff. by part as discussed in formula (2)

⇒ 

Since,  & 

⇒ 

⇒        (∵ y = (xx)x)

Put x = 1 in above equation

⇒  = 1 + 2 × 1 log 1

⇒  = 1          (∵ log 1 = 0)

∴  at x = 1 is 1.

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