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.
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