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Homework Help: Finding the differential of the function

  1. Oct 28, 2006 #1
    We just finished linear approximations and differentials in our course and was hoping I could get some feedback on this question of mine. The following question I have is asking to find the differential of the function:

    y= cos(pi)x

    We know from the formula

    dy = f'(x)*dx

    However, we are supposed to use our product rule here to determine the derivative of cos(pi)x.

    I came up with -(pi)sin(pi)x but I'm not sure how to solve from that using the rule. The rest is fairly straight forward because it's just putting it back into the formula. Thanks guys for any help.
  2. jcsd
  3. Oct 28, 2006 #2
    [tex] y = \cos(\pi) x [/tex]

    [tex] \frac{dy}{dx} = \cos (\pi) [/tex]

    [tex] y = \cos(\pi x) [/tex]

    [tex] \frac{dy}{dx} = -\pi \sin \pi x [/tex]. You are using the chain rule.

    [tex] dy = -\pi \sin \pi x \ dx [/tex]
    Last edited: Oct 28, 2006
  4. Oct 28, 2006 #3
    I think loadsy means cos(pi x).
  5. Oct 28, 2006 #4
    In which case it would simply be:

    y = cos(pi*x)
    dy/dx = -pi*sin(pi*x)

    Differential form would be:

    dy = -pi*sin(pi*x)*dx

    Forget formulas like that. Just remember to multiply both sides by dx after taking the derivative of the function.
  6. Oct 28, 2006 #5
    Actually it's just written as y = cos pi x. Which I think is basically the same as [tex] y = \cos\pi x [/tex]

    But I understand what you are saying anyways. :D
    Last edited: Oct 28, 2006
  7. Oct 28, 2006 #6
    Yeah, so go with my answer.
  8. Oct 28, 2006 #7
    Gotcha! Alright thanks for the help guys. The question was easier than I made it out to be.
  9. Oct 28, 2006 #8
    Alright I hope you guys don't mind, I just have another question to ask you, but it should go by fast because I know the steps in solving it anyways.

    This is under the "Mean Value Theorem" section, and asks:

    Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers cs that satisfy the conclusion of the Mean Value Theorem.

    f(x) = x / x+2 [1,4]

    We know the following:

    f(x) is differentiable on (1,4) and that (1,4) is part of domain of definition of f'(x).
    f(x) is continuous on [1,4] because it is a rational function.

    Then we have to solve for f'(x) which I found to be:

    f'(x) = x+2*1 - x*2 / (x+2)^2 = -2x / (x+2) but I'm not sure if this is the right derivative for this case. Once I have that the rest I can do to solve for c.

    Because from there you'd solve using the formula f'c = f(4)-f(1) / 4-1 = 1/9 after finding out:
    f(4) = 4/4+2 = 2/3 and f(1) = 1/1+2 = 1/3

    So in summation: all I'm looking to find the answer of, is the derivative of x / x+2
    Last edited: Oct 28, 2006
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