Use the logarithmic differentiation of the following. f(x)=x^{x}

Yulia

Yulia

Answered question

2020-12-06

Use the logarithmic differentiation of the following. f(x)=xx

Answer & Explanation

averes8

averes8

Skilled2020-12-07Added 92 answers

f(x)=xx=>ddxxx=
Apply exponent rule: ab=ebln(a)xx=exln(x)=ddxexln(x)
Apply the chain rule:
df(u)dx)=dfdu×dudxf=eu,u=xln(x)=ddu(eu)ddxxln(x)=eu(ln(x)+1)=exln(x)(ln(x)+1)=xx(ln(x)+1)
ddx(xx)=xx(ln(x)+1)

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