# Derivatives of Logarithmic Functions -Fractions-

• Oct 23rd 2008, 06:44 PM
GirouxCalder
Derivatives of Logarithmic Functions -Fractions-
Hello everyone.

I got a question regarding the derivatives of a log but in a fraction. Here is an example:

3x^2 / ln(x)

This isnt a homework problem but im studying for a quiz tomorrow and want to be ready for anything. Would I use the quotient rule for this problem?

Also another quick question. Its kind of stupid so sorry.
But in this question how do i get from step 1 to step 2.

x= 1
y = sin (7 ln(x))
(derivative) dx/dy y= cos(7ln(x)) (7/x)

Step 1: y(1) = cos (7ln(1)) (7/1)
Step 2: y(1) = cos (0) (7)

Im guessing ln (1) = 0 ?

thanks for your help
• Oct 23rd 2008, 06:48 PM
Jameson
Quote:

Originally Posted by GirouxCalder
Hello everyone.

I got a question regarding the derivatives of a log but in a fraction. Here is an example:

3x^2 / ln(x)

This isnt a homework problem but im studying for a quiz tomorrow and want to be ready for anything. Would I use the quotient rule for this problem?

Hi GirouxCalder,

Yes for this you would use the quotient rule. Just take note that $\frac{d}{dx} \ln(x) = \frac{1}{x}$. So you'll end up with a fraction in your larger fraction, which just means a little tricky simplification. I think these problems are missed more on the algebra used to clean up the answer than the actual derivative.

Jameson
• Oct 23rd 2008, 06:50 PM
Jameson
Quote:

Originally Posted by GirouxCalder
Also another quick question. Its kind of stupid so sorry.
But in this question how do i get from step 1 to step 2.

x= 1
y = sin (7 ln(x))
(derivative) dx/dy y= cos(7ln(x)) (7/x)

Step 1: y(1) = cos (7ln(1)) (7/1)
Step 2: y(1) = cos (0) (7)

Im guessing ln (1) = 0 ?

Yes ln(1)=0.

Remember that if $\ln(a)=b$, this is equivalent to saying $e^b=a$. Here a=1, so b must be zero for this statement to be true.
• Oct 23rd 2008, 07:33 PM
GirouxCalder
Thanks, I still just cant figure these problems out though... I think if i can get one i will be able to get the rest. Mind helping me?

heres a example:

4 / (ln(x))^2

for some reason i get the concept but cant understand how to solve it....
• Oct 23rd 2008, 07:41 PM
Jameson
Quote:

Originally Posted by GirouxCalder
Thanks, I still just cant figure these problems out though... I think if i can get one i will be able to get the rest. Mind helping me?

heres a example:

4 / (ln(x))^2

for some reason i get the concept but cant understand how to solve it....

Well this one can be done just as easily using the power and chain-rule, or by the quotient rule. Since you asked about the quotient rule, I'll do it that way.

$\frac{d}{dx} \frac{4}{(\ln x)^2} = \frac{[(\ln x)^2 *0] -(4)[2(\ln(x))*(\frac{1}{x})]}{(\ln x)^4}$

Simplifying,

$... = - \frac{8}{x(\ln x)^3}$