Help with Logarithmic Differentiation

[communitycoll]

Homework Statement

y = [2x + 1]^5 * [(x^4) - 3]^6

Homework Equations

I take the derivative of the natural log of both sides:

(y' / y) = [(10 ln(2x + 1)^4) / (2x + 1)] + [(24x^3 ln(x^4 - 3)^5) / (x^4 - 3)]

then I multiply both sides by the original function:

y' = [((10 ln(2x + 1)^4) / (2x + 1)) + ((24x^3 ln(x^4 - 3)^5) / (x^4 - 3))] * [(2x + 1)^5 * ((x^4) - 3)^6]

My book, however, says the answer is:

y' = [(10 / (2x + 1)) + (24x^3 / (x^4 - 3))] * [(2x + 1)^5 * ((x^4) - 3)^6]

^ i.e., the same thing without the logs.

Could someone please explain what I did wrong, or where I might have gone wrong?

The Attempt at a Solution

What you see above.

 

[SammyS]

Homework Statement

y = [2x + 1]^5 * [(x^4) - 3]^6

Homework Equations

I take the derivative of the natural log of both sides:

(y' / y) = [(10 ln(2x + 1)^4) / (2x + 1)] + [(24x^3 ln(x^4 - 3)^5) / (x^4 - 3)]

then I multiply both sides by the original function:

y' = [((10 ln(2x + 1)^4) / (2x + 1)) + ((24x^3 ln(x^4 - 3)^5) / (x^4 - 3))] * [(2x + 1)^5 * ((x^4) - 3)^6]

My book, however, says the answer is:

y' = [(10 / (2x + 1)) + (24x^3 / (x^4 - 3))] * [(2x + 1)^5 * ((x^4) - 3)^6]

^ i.e., the same thing without the logs.

Could someone please explain what I did wrong, or where I might have gone wrong?

The Attempt at a Solution

What you see above.

It's helpful to do the problem in steps.

What did you get for ln(y) ?

Then, what did you get for the derivative of that result?

 

[communitycoll]

It's helpful to do the problem in steps.

What did you get for ln(y) ?

Then, what did you get for the derivative of that result?

I got ln(y) for ln(y), I'm not sure I understand the first question.

The derivative of that I got:

(1 / y)(y')

 

[Villyer]

I think that you took your derivitives wrong.

I see that you broke it up into the sum of two logs, but remember that they are of the form ln(u^n) and not of the form ln(u)^n.

 

[communitycoll]

I think that you took your derivitives wrong.

I see that you broke it up into the sum of two logs, but remember that they are of the form ln(u^n) and not of the form ln(u)^n.

I use chain rule on each of those logs; should I not?

 

[communitycoll]

Ah, forget I asked anything. I understand what I did wrong. Thanks anyway.

 

[SammyS]

I got ln(y) for ln(y), I'm not sure I understand the first question.
...

If y = [2x + 1]⁵ * [(x^4) - 3]⁶, then ln(y) = __?__ .

 

[Chestermiller]

Is there a reason you didn't apply the relationship for the derivative of the product of two functions? Did the question constrain you to use logarithmic differentiation?