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Question

Differentiate the following functions w.r.t. x:
xxx

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Solution

We have,

(xx)x


Let,

y=(xx)x=xxx


Taking log both side and we get,

logy=log(xxx)

logy=xxlogx


Again taking log both side and we get,

loglogy=log(xxlogx)

loglogy=logxx+loglogx

loglogy=xlogx+loglogx


On differentiation and we get,

1logy1ydydx=xddxlogx+logxdxdx+1xlogx

1ylogydydx=x×1x+logx+1xlogx

dydx=ylogy(1+logx+1xlogx)

dydx=xxxlogxx(1+logx+1xlogx)


Hence, this is the answer.


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