1. Logarithmic Differentiation

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. Originally Posted by KK88
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.

3. Originally Posted by KK88
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

4. Originally Posted by KK88
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