Logarithmic Differentiation

May 2010
6
1
Hi I am having trouble solving a couple of problems involving logarithmic differentiation.

1.Find
if


2.If
, find
.

3.Let

Determine the derivative
at the point
.

4. If
, find
.

Any tips on how to do these? Thanks.
 
Feb 2010
1,036
386
Dirty South
Hi I am having trouble solving a couple of problems involving logarithmic differentiation.

1.Find
if


2.If
, find
.

3.Let

Determine the derivative
at the point
.

4. If
, find
.

Any tips on how to do these? Thanks.
For the first question, use the chain rule by solving:

\(\displaystyle \frac{\mbox{d log(u)}}{\mbox{du}} \frac{\mbox{du}}{\mbox{dx}}\) where \(\displaystyle u = \sqrt{\frac{4x+8}{5x+7}}\)

and differentiate..

For further assistance, show your work on this and the other problems on where you are getting stuck.
 
Apr 2010
384
153
Canada
Hi I am having trouble solving a couple of problems involving logarithmic differentiation.





1.Find
if






2.If
, find
.​






3.Let







Determine the derivative
at the point
.​



4. If
, find
.​



Any tips on how to do these? Thanks.​
2:

The first part of this is easy, but the derivative of \(\displaystyle x^x \) is not, so let us go through that

\(\displaystyle y = x^x \)

\(\displaystyle lny = xlnx \)

\(\displaystyle \frac{1}{y} y` = lnx + 1 \)

\(\displaystyle y` = x^x lnx + x^x \)

Thus,

\(\displaystyle F(x) = 4sinx + 3x^x \)

\(\displaystyle F`(x) = 4cosx + 3x^x( lnx + 1) \)

3:

For \(\displaystyle y= ln(x^2 + y^2) \)

\(\displaystyle y` = \frac{1}{x^2 + y^2} (x^2+y^2)` = \frac{1}{x^2 + y^2} (2x + 2y y` ) \)

Bring y prime over to one side and factor it out,

\(\displaystyle y`[ 1 - \frac{2y}{x^2 + y^2} ] = \frac{1}{x^2 + y^2} (2x) \)

\(\displaystyle y` = \frac { \frac{2x}{x^2 + y^2} }{ 1 - \frac{2y}{x^2 + y^2} } \)

Sub in the point (1,0) to find the value.

4:

is a repeat of 2
 
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Jul 2007
894
298
New Orleans
Hi I am having trouble solving a couple of problems involving logarithmic differentiation.






1.Find
if







2.If
, find
.​







3.Let








Determine the derivative
at the point
.​



4. If
, find
.​



Any tips on how to do these? Thanks.​

Here is abother way to approach the first problem. Use your log rules to seperate the the experession.

\(\displaystyle y = \frac{1}{2}\bigg(\ln{(4x+8)} - \ln{(5x-7)}\bigg)\)

\(\displaystyle y' = \frac{1}{2}\bigg(\frac{4}{4x+8}-\frac{5}{5x-7}\bigg)\)

now just simplify
 
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