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
explanation: The document is a series of posts from a user named KK88 who is seeking help with logarithmic differentiation problems. The first post lists four problems and asks for tips on how to solve them. The subsequent posts provide solutions or alternative approaches to the problems.
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distilabel_metadata.content: 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
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a series of posts from a user named KK88 who is seeking help with logarithmic differentiation problems. The first post lists four problems and asks for tips on how to solve them. The subsequent posts provide solutions or alternative approaches to the problems.
  
  Label: multiple-question-and-answer