How to compute the derivative of sqrt x ^(sqrt x) I know have the final answer and know I need to use the natural log but I'm confused about why that is.

Alice Chen

Alice Chen

Answered question

2022-11-11

How to compute the derivative of x x ?
I know have the final answer and know I need to use the natural log but I'm confused about why that is.
Could someone walk through it step by step?

Answer & Explanation

Ryan Davies

Ryan Davies

Beginner2022-11-12Added 18 answers

Let y = x x . Then ln ( y ) = x ln ( x ) = 1 2 x ln ( x ). So,
d d x ln ( y ) = d d x ( 1 2 x ln ( x ) )
1 y d y d x = 1 2 ( x x + 1 2 x ln ( x ) )
So,
d y d x = y 1 2 ( x x + 1 2 x ln ( x ) ) = 1 2 x x ( x x + 1 2 x ln ( x ) )
Jefferson Booth

Jefferson Booth

Beginner2022-11-13Added 4 answers

Notice that x x = x 1 2 x . Applying the chain rule yields the derivative:
( 1 2 x ) x 1 2 x 1 + ( ln x ) 1 2 1 2 x x 1 2 x .

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