Derivative of a log function

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

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

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Originally Posted by radioheadfan

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

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

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Let $\displaystyle g(x)= \ln f(x)$ represent the given fuction. The derivitive is then $\displaystyle [1/f(x)]\cdot f^\prime (x).$ Use the Quotient Rule or Product Rule for the last bit.

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Originally Posted by radioheadfan

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

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

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


Edit: By the way, you better note that $\displaystyle \ln x^2 = 2 \ln |x|$.

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Originally Posted by mr fantastic

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

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Or that...

I used the log rules and got 2/3

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Originally Posted by radioheadfan

I used the log rules and got 2/3

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What happened to all the x's?

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Originally Posted by radioheadfan

I used the log rules and got 2/3

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Don't think that's right...

Using the log rules, we get

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

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

Now differentiate...