Help with differentiation of natural logarithm Find <mspace width="thickmathspace" /> <mstyl

racodelitusmn

racodelitusmn

Answered question

2022-07-04

Help with differentiation of natural logarithm
Find d y d x given y = ln ( 8 x ) 8 x
The answer is 1 ln ( 8 x ) 8 x 2
Can you show the process of how this is worked?
Thanks.

Answer & Explanation

Zane Barry

Zane Barry

Beginner2022-07-05Added 5 answers

Here we can use the quotient rule and the chain rule:
Quotient rule: y = f ( x ) g ( x ) , then ((1) quotient rule) d y d x = f ( x ) g ( x ) f ( x ) g ( x ) g 2 ( x )
f ( x ) = ln ( 8 x ) , f ( x ) = 1 8 x d d x ( 8 x ) = 8 8 x = 1 x by chain rule
g ( x ) = 8 x , g ( x ) = 8.
Now, just substitute each of f ( x ) , g ( x ) , f ( x ) , g ( x ) , and g 2 ( x ) = ( 8 x ) 2 into (1), simplify, and you're done!
d y d x = ( 8 x ) 1 x ( ln ( 8 x ) ) 8 64 x 2 = ( 1 ln ( 8 x ) ) 8 x 2
babyagelesszj

babyagelesszj

Beginner2022-07-06Added 7 answers

This is just the quotient rule:
1 8 [ x 1 x 1 ln ( 8 x ) x 2 ] .

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