Thursday, November 09, 2006

Chapter 3 Test Question 9

Differentiate the function:

y = ln( ((6x^9 + 6x)^2(5x^3 + 9))/(5x^9 + 5x)^(1/2) )

Sorry about the super-primative notation. Equation editor is suddenly nonexistent on my computer. But okie dokie...

First, use the laws of logarithms to make this messy thing simpler to deal with:
  • Reminder: When one number (or expression) in the ln is divided by a number, you can subtract the ln of the bottom by the ln of the top. When one number (or expression) in the ln is multiplied by another, you can add the ln of one expression to the ln of the other. Finally, when a number is in the exponent of an ln, you can multiply the entire ln by that number. Thus...
  1. y = ln( (6x^9 + 6x)^2 (5x^3 + 9) ) - ln( (5x^9 + 5x)^(1/2) )
  2. y = ln( (6x^9 + 6x)^2) + ln(5x^3 +9) - ln( (5x^9 + 5x)^(1/2) )
  3. y = 2ln(6x^9 + 6x) + ln(5x^3 +9) - (1/2)ln(5x^9 +5x)
  • THEN, now that you have a nice, straightforward (albeit rather ugly) equation, you can take the derivative, keeping in mind that when dealing with ln, y' = (g'(x))/(g(x)). So, let's break it up...
  1. g(x) = (6x^9 + 6x).....SO.....g'(x) = (54x^8 + 6)
  2. h(x) = (5x^3 + 9).....SO.....h'(x) = (15x^2)
  3. k(x) = (5x^9 + 5x).....SO.....k'(x) = (45x^8 + 5)

Put that all together into the equation -- g'(x)/g(x) -- then factor in the coefficients using the Chain Rule, and what do you get?

y' = 2( (54x^8 + 6) / (6x^9 + 6x) ) + ( (15x^2) / (5x^3 + 9) ) - (1/2)( (45x^8 + 5) / (5x^9 + 5x) )

Yay for being finished!

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