# Thread: Derivative of a log function

1. ## Derivative of a log function

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

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

2. 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.

3. 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 $\ln x^2 = 2 \ln |x|$.

4. Originally Posted by mr fantastic
Use the usual log rules to get an expression that is simpler to differentiate.
Or that...

5. I used the log rules and got 2/3

6. Originally Posted by radioheadfan
I used the log rules and got 2/3
What happened to all the x's?

7. 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...