Differentiate, please: y=x^{x}.

Julia White

Julia White

Answered question

2021-12-09

Differentiate, please: y=xx.

Answer & Explanation

Virginia Palmer

Virginia Palmer

Beginner2021-12-10Added 27 answers

Use logarithmic diferentiation:
lny=lnxx
Diferentiaate both sides with respect to x:
1ydydx=lnx+x1x
1ydydx=lnx+1
Multiply by y:
dydx=y(lnx+1)
Now y=xx, thus,
dydx=xx(lnx+1)
Jeremy Merritt

Jeremy Merritt

Beginner2021-12-11Added 31 answers

Apply exponent rule:
ab=ebln(a)
xx=exln(x)
=(ddx)exln(x)
Now, apply the Chain Rule:
=(ddx)exln(x)=exln(x)ddx(xln(x))
=exln(x)(ln(x)+1)
xx(ln(x)+1)

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