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Derivative Question- Chain Rule

  1. May 16, 2010 #1
    1. The problem statement, all variables and given/known data

    The derivative of the function
    h(x) = sin((x2 + 1)2)

    2. Relevant equations

    Chain Rule

    3. The attempt at a solution

    h(x) = sin((x2 + 1)2)

    f(u) = sinu^2, f'(u)= 2ucosu^2
    g(x) = x^2+1 g(x)= 2x

    I get lost putting this back together but:

    2(sinu^2)[cos(sinu^2)^2](2x) ?
  2. jcsd
  3. May 16, 2010 #2
    Are you trying to differentiate h(x) = sin [(x2 + 1)2] ? I'm currently having a hard time following your procedure (with the switching around of the h's and f's and u's and g's).
    Last edited: May 16, 2010
  4. May 16, 2010 #3


    Staff: Mentor

    This apparently is h(x) = sin((x2 + 1)2).
    My guess is that your are looking at a formula for the chain rule as h(x) = f(g(x)).
    In your problem, f(u) = sin(u), not sin2(u), so f'(u) = cos(u).

    g(x) = ((x2 + 1)2), so g'(x) = ??

  5. May 17, 2010 #4
    Sorry, I was a bit out of it last night.

    h(x) = sin(x^2+1)^2

    I have been taught to break this into f(u) and g(x) to apply chain rule


    f(u) = sin(u) f'(u)=cos(u)
    g(x) = (x^2+1)^2 g'(x) = 4x(x^2+1)

    h'(x) = cos((x^2+1)^2)4x(x^2+1)
  6. May 17, 2010 #5
    Yep that seems right.

    Just another note, I understand what you're doing, but I think if you end up confusing yourself by changing variables then it might not be worth it. I always thought of the chain rule as "Derivative of the outside function * derivative of inside function."
  7. May 17, 2010 #6
    I conceptually could not grasp the chain rule last night. Today it "clicked" when I thought about it in those terms.
  8. May 18, 2010 #7


    Staff: Mentor

    Part of the difficulty with this problem, from our side, was trying to understand what the function was.

    In your first post, you had h(x) = sin((x2 + 1)2).

    Later, you wrote this as h(x) = sin(x^2+1)^2. This is still somewhat ambiguous, as it is not clear exactly what is being squared. From you derivative, it seems to be this:
    h(x) = sin((x^2+1)^2).

    In the latter form it is clear that x --> x^2 + 1 --> (x^2 + 1)^2 --> sin((x^2 + 1)^2).

    Part of being able to get help in this or other forums or other resources is being able to write your question clearly and unambiguously.
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