Emmanuel Pace

Emmanuel Pace

Answered

2022-07-20

y = ( 3 x 2 + 2 ) l n x
The answer to this is:
d y d x = ( 3 x 2 + 2 ) l n x ( 1 x l n ( 3 x 2 + 2 ) + 6 x l n x 3 x 2 + 2 )
What I'm coming up with is:
d y d x = ( 3 x 2 + 2 ) l n x ( 1 x l n ( 3 x 2 + 2 ) + 6 x 3 x 2 + 2 )
What I'm not understanding is where the 6 x l n x 3 x 2 + 2 comes from, if anyone could explain this I'd really appreciate it.

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Answer & Explanation

umgangistbf

umgangistbf

Expert

2022-07-21Added 12 answers

Using logarithmic differentiation, that is,
ln ( y ) = ln ( x ) ln ( 3 x 2 + 2 ) .
Now, when you take the derivative, the left-hand side becomes
y y .
Use the product rule on the right-hand side, and don't forget y = ( 3 x 2 + 2 ) ln ( x ) . Thus, altogether you'll get
y y = 1 x ln ( 3 x 2 + 2 ) + ln ( x ) 3 x 2 + 2 ( 6 x ) .
Above, the 6x comes from the chain rule. Multiply both sides by y = ( 3 x 2 + 1 ) ln ( x ) and we get
y = ( 3 x 2 + 2 ) ln ( x ) ( 1 x ln ( 3 x 2 + 2 ) + 6 x ln ( x ) 3 x 2 + 2 ) .
We can perhaps go a little further in simplification to get
y = ( 3 x 2 + 2 ) ln ( x ) + 1 ( 1 x + 6 x ln ( x ) ( 3 x 2 + 2 ) 2 ) .
Here, I have factored a ( 3 x 2 + 2 ) from each term, thus the second term will get a square in the denominator, and the factor in front of the fraction gets an extra +1 in the numerator.

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