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If y=x^(x^(2)), then (dy)/(dx) equals...

If `y=x^(x^(2)),` then `(dy)/(dx)` equals

A

`2logx.x^(2)`

B

`(2logx+1).x^(x^(2))`

C

`(2logx+1).x^(x^(2))+1`

D

`x^(x^(2)+1).(log(ex^(2)))`

Text Solution

Verified by Experts

The correct Answer is:
D
Taking log both sides
`logy = logx^(x^2)`
`logy = x^2logx` [Using product rule]
`(1/y)*(dy/dx) = x^2 (1/x) + 2x*logx`
`= x + 2x logx`
`=> (1/x^(x^2))dy/dx = x + 2x*logx`
`(dy/dx) = x^(x^2)( x + 2x*logx)`
= `x^(x^2)*x(1 + 2logx)`
`= x^((x^2)+1)(1 + 2logx)`
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