# Logarithmic Differentiation

#### 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.

#### harish21

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.

#### AllanCuz

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

KK88

#### 11rdc11

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

HallsofIvy