JEE Main 2017 · Mathematics · Differentiation and Application of Derivatives · medium

For the function y=x^x defined on x>0, the value of its derivative at the point x=1 is

Single correct option

For the function

y=x^x

defined on

x>0

, the value of its derivative at the point

x=1

Show answer and solution

Answer: (B) $1$

Take logarithms:

\ln y=x\ln x

. Differentiate to get

\dfrac1y\dfrac{dy}{dx}=\ln x+1

, so

\dfrac{dy}{dx}=x^x(1+\ln x)

. At

x=1

this is

1\cdot(1+0)=1

. Treating

x^x

as a power function would wrongly give

x\cdot x^{x-1}=1

only by coincidence at this point, so the logarithmic method is the safe one.

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