Hi I am having trouble solving a couple of problems involving logarithmic differentiation.

1.Find

if

3.Let

Determine the derivative

at the point

.

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