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Textbook Example

  1. Sep 27, 2013 #1
    1. The problem statement, all variables and given/known data

    May I please have help understanding this example problem in my textbook.

    I am working on the chain rule for finding derivatives in Calculus. What I am currently having trouble with is factoring. My textbook gives the follwing example:

    f'(x) =(3x - 5)4(7 - x)10

    f'(x) = (3x - 5)4 x 10(7 - x)9(-1) + (7 - x)104(3x - 5)3(3)

    = -10(3x - 5)4(7 - x)9 + (7 - x)1012(3x - 5)3

    Here it says 2(3x - 5)3(7 - x)9 is factored out.

    = 2(3x - 5)3(7 - x)9[-5(3x - 5) + 6(7 - x)]

    =2(3x - 5)3(7 - x)9(-15x + 25 + 42 - 6x)

    And it continues onto simplification.

    The area I have trouble with is when 2(3x - 5)3(7 - x)9 is factored out.

    I am not fully understanding why

    = 2(3x - 5)3(7 - x)9[-5(3x - 5) + 6(7 - x)]

    is the result. How was it derived?

    Thank you

    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 27, 2013 #2

    DrClaude

    User Avatar

    Staff: Mentor

    Maybe it will be simpler for you to understand if we rewrite
    as
    $$
    (2) (-5) a^4 b^9 + (2) (6) b^{10} a^3
    $$
    A factor of 2 is common to both terms, and you can factor out lowest power of ##a## and of ##b##, that is ##2 a^3 b^9##. Divide the first term by this factor,
    $$
    \frac{(2)(-5) a^4 b^9 }{2 a^3 b^9} = -5 a,
    $$
    and do the same with the second term. You get
    $$
    (2) (-5) a^4 b^9 + (2)(6) b^{10} a^3 = 2 a^3 b^9 \left( -5 a + 6 b \right)
    $$
     
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