# Thread: Derivatives of Logarithmic Functions -Fractions-

1. ## 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 ?

2. 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 $\displaystyle \frac{d}{dx} \ln(x) = \frac{1}{x}$. 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

3. 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 $\displaystyle \ln(a)=b$, this is equivalent to saying $\displaystyle e^b=a$. Here a=1, so b must be zero for this statement to be true.

4. 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....

5. 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.

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

Simplifying,

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