How do you differentiate $y=x^x$ ?

Question

75296 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-03-05T22:05:26

You can use logarithmic differentiation

Take the natural logarithm of both sides

$lny=lnx^x$

Now using properties of logarithms, rewrite the right hand side

$lny=xlnx$

Differentiate both sides with respect to $x$
Use the product rule on the right side

$1/ydy/dx=lnx+x1/x$

$1/ydy/dx=lnx+1$

Multiply both sides by $y$

$dy/dx=y(lnx+1)$

Now $y=x^x$ so we can write

$dy/dx=x^x(lnx+1)$

How do you differentiate y=x^xy=x^x?