text: 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