First and Second Derivatives

  1. 1. The problem statement, all variables and given/known data
    Just trying to find the first and second derivatives.

    X^2/(X^2-16)

    1+X/1-X

    X^3(X-2)^2

    2. Relevant equations
    Quotient Rule/Power Rule/Chain Rule


    3. The attempt at a solution
     
  2. jcsd
  3. For the first one, what is the derivative of f(x)/g(x)? In general, not for this specific function.
     
  4. (g(x)*f'(x)-f(x)*g'(x))/g(x)^2
     
  5. okay, so if f(x) = x^2 and g(x) = x^-2, what is the derivative of f(x)/g(x)?
     
  6. f(x) = x^2 and g(x) = x^-2
    x^2/x^-2
    ((x^-2)(2x)-(-2x)(x^2))/ (x^2)^2
     
  7. I mistyped (that g(x) should have been g(x)=x^2 - 16 [not sure how I messed that one up]), but you seem to know what you're doing. What specific question do you have?
     
  8. I'm mostly having trouble with the second derivatives.
     
  9. What do you have so far? Where are you getting stuck?
     
  10. Well here's what I've got, I think they're right but I'm not sure.

    f(x)=X^2/(x^2-16)
    (X^2-16)(2X)-(X^2)(2X)/(X^2-16)^2
    f'(x)=-32X^2/(X^2-16)^2
    (X^4-32X^2+256)(-64X)-(-32X^2)(4X^3-64X)/(X^4-32X^2+256)^2
    -64x^5+2048X^3-16384X+128X^5-2048X^3
    f''(x)=64X^5-16384X/(X^4-32X^2+256)^2

    f(x)=1+x/1-X
    (1-X)(1)-(1+X)(-1)/(1-X)^2
    f'(x)=2-2X/(1-X)^2
    (X^2-2X+1)(-2)-(2X-2)(2-2X)/(X^2-2X+1)^2
    f''(x)=2X^2+4X+2/(X^2-2X+1)^2

    f(x)=X^3(X-2)^2

    (X^3)(2X-4)+(3X^2)(X-2)^2
    2X^4-4X^3+3X^4-12X^3+12X^2
    f'(x)=5X^4-16X^3+12X^2
    5X^4-16^3+12X^2
    f''(x)=20X^3-48X^2+24X
     
  11. Am I doing these correctly?
     
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