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The chain rule

  1. Oct 12, 2011 #1
    1. The problem statement, all variables and given/known data

    A curve has equation [tex]y=(x^2+1)^4 + 2(x^2+1)^3[/tex]. Show that [tex]\frac{dy}{dx}=4x(x^2+1)^2(2x^2+5)[/tex].

    2. Relevant equations

    [tex] \frac{dy}{dx}=(\frac{dy}{du}\times \frac{du}{dx})+(\frac{dy}{dv} \times \frac{dv}{dx})[/tex]

    3. The attempt at a solution

    [tex]y=(x^2+1)^4 + 2(x^2+1)^3[/tex]

    [tex]let u = (x^2+1)^4 and v=x^2+1 so that y=u^4+v^3[/tex]

    [tex]\frac{dy}{du}=4u^3=4(x^2+1)^3 and \frac{du}{dx}=2x[/tex]

    [tex]\frac{dy}{dv}=3v^2=3(x^2+1)^2 and \frac{dv}{dx}=2x[/tex]

    [tex]\therefore \frac{dy}{dx}=(\frac{dy}{du}\times \frac{du}{dx})+(\frac{dy}{dv} \times \frac{dv}{dx})=8x(x^2+1)^3+6x(x^2+1)^2[/tex]

    Why am I not getting the answer [tex]4x(x^2+1)^2(2x^2+5)[/tex]?
     
  2. jcsd
  3. Oct 12, 2011 #2

    WELL YOU HAVE TO FACTOR IT. 8x(x^2+1)^3+12x(x^2+1)^2
    IS THE SAME AS 4X [2(X^2+1)^3 +3(X^2+1)^2]....KEEP DOING IT..AT THE END YOU WILL HAVE THE ANSWER

    BY THE WAY IT SHOULD BE 12 X NOT 6X
     
  4. Oct 13, 2011 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    I would start by writing
    [tex]y=(x^2+1)^4+2(x^2+1)^3= (x^2+ 1)(x^2+ 1)^3+ 2(x^2+ 1)^3= (x^2+ 1)^3(x^2+ 3)[/tex].

    Now,
    [tex]dy/dx= 3(x^2+ 1)^2(2x)(x^2+ 3)+ (x^2+ 1)^3(2x)[/tex]
     
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