Derivative of a log function

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August 4th 2009, 08:42 PM
radioheadfan

Derivative of a log function

Normally I would know how to get a simple log derivative, but this problem is much more difficult.

$ln \frac{x^2\sqrt{4x+1}}{(x^3+5)^3}$
August 4th 2009, 08:47 PM
AlephZero

Quote:

Originally Posted by radioheadfan

Normally I would know how to get a simple log derivative, but this problem is much more difficult.

$ln \frac{x^2\sqrt{4x+1}}{(x^3+5)^3}$

Let $g(x)= \ln f(x)$ represent the given fuction. The derivitive is then $[1/f(x)]\cdot f^\prime (x).$ Use the Quotient Rule or Product Rule for the last bit.
August 4th 2009, 08:48 PM
mr fantastic

Quote:

Originally Posted by radioheadfan

Normally I would know how to get a simple log derivative, but this problem is much more difficult.

$ln \frac{x^2\sqrt{4x+1}}{(x^3+5)^3}$

Use the usual log rules to get an expression that is simpler to differentiate.

Edit: By the way, you better note that $\ln x^2 = 2 \ln |x|$ .
August 4th 2009, 08:49 PM
AlephZero

Quote:

Originally Posted by mr fantastic

Use the usual log rules to get an expression that is simpler to differentiate.

Or that...
August 4th 2009, 09:54 PM
radioheadfan

I used the log rules and got 2/3
August 4th 2009, 09:56 PM
mr fantastic

Quote:

Originally Posted by radioheadfan

I used the log rules and got 2/3

What happened to all the x's?
August 4th 2009, 10:04 PM
AlephZero

Quote:

Originally Posted by radioheadfan

I used the log rules and got 2/3

Don't think that's right...

Using the log rules, we get

$\ln x^2 + \ln\sqrt{4x+1} - \ln[(x^3+5)^3]$

$=2\ln|x| +\frac{1}{2}\ln(4x+1) -3\ln(x^3+5).$

Now differentiate...

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