derivatives of logarithmic functions

[michi]

Hi all. I'm having trouble with a couple of problems dealing with logarithmic functions.

my first problem deals with the function: -3ln[cosx]

I have to find the second derivative and I keep getting 3(sec^2)x. Why is that wrong???? I'm doing internet based homework and it doesn't want to take it. Am I missing something?

My other problem is, f(x)=ln[sqrt(2x-6/5x+6)].

For this one, I know I simplify it to ln (2x-6/5x+6)^1/2 then I pull the 1/2 to the front and get

1/2 ln [2x-6/5x+6]

Then I have 1/2[ln(2x-6)-ln(3x-2)] because I split the ln up.

And then when I multiply the 1/2 to both of these functions what do I do. I don't know how to correctly simplify the answer further. I'm a bit confused because my ta multiplied it only times the numerator. Is that what I'm supposed to do? It gives me a wrong answer when I input it.

Thanks in advanced for help.

 

[Unco]

my first problem deals with the function: -3ln[cosx]

I have to find the second derivative and I keep getting 3(sec^2)x. Why is that wrong???? I'm doing internet based homework and it doesn't want to take it. Am I missing something?

Show your work and be careful with the chain rule. Edit: my bad; >second< derivative.

My other problem is, f(x)=ln[sqrt(2x-6/5x+6)].

For this one, I know I simplify it to ln (2x-6/5x+6)^1/2 then I pull the 1/2 to the front and get

1/2 ln [2x-6/5x+6]

Then I have 1/2[ln(2x-6)-ln(3x-2)] because I split the ln up.

What happened there? (Nice explanations, by the way.)

1/2 * ln((2x - 6)/(5x + 6)) = 1/2*ln(2x - 6) - 1/2*ln(5x + 6), right?

Now you can differentiate term by term (by hand, I suggest).

 

[Gene]

It might just be your typing. Try
y'' = 3*sec^2(x)
The math looks good.

Watch your typing here too. It should be
f(x)=ln[sqrt((2x-6)/(5x+6))]
You are all right till
1/2[ln(2x-6)-ln(3x-2)]
It should be
1/2[ln(2x-6)-ln(5x+6)] =
1/2[ln(2x-6)]-1/2[ln(5x+6)]
Spliting it up doesn't change the values.

I think you missed something with TA's work.
sqrt[(2x-6)/(5x+6)] =
sqrt(2x-6)/sqrt(5x+6)
If you look at it that way you can see that both numerator and denominator have the 1/2 multiplier with the ln.

Simplification is a shakey concept. It can mean different things. Usually you don't want sqrt in the denominator. You might try
sqrt[(2x-6)(5x+6)]/(5x+6) at some point.