Find the derivative of a function ln ⁡ x x .

liachta6VW

liachta6VW

Answered

2022-11-25

Find the derivative of a function ln x x .

Answer & Explanation

Aleah Rowe

Aleah Rowe

Expert

2022-11-26Added 15 answers

The quotient rule states that:
( f g ) = f g f g g 2
where the apostrophe ( ) means "the derivative of".
In this example, we let f = ln x and g = x
This allows us to rewrite the function as:
( ( d d x ln x ) x ln x ( ( d d x ) x ) x 2
Other calculus rules tell us that the derivative of ln x is always 1 x . As a result, the first half of our numerator becomes x ( 1 x ), Which neatly simplifies to 1. This gives:
1 ln x ( ( d d x ) x ) x 2
Now we can use the power rule, which states that ( d d x ) x n = n x n 1 . Since we know that nn in this case is 1 (because x has no exponent), this becomes 1 × x 1 1 , which yields a value of 1. This gives us our final answer:
1 ln x x 2

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