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Differential Quotient

  1. Apr 8, 2009 #1
    I don't take a Calculus class(I'm learning on my own), but I'm just curious as to what are the steps to solving the following equations.

    1. The problem statement, all variables and given/known data



    These are the following problems that I'm having trouble solving.

    y(x) = cos(x)

    y(x) = [tex]\sqrt{x}[/tex]

    y(x) = sin(x)

    y(x) = x[tex]^{n}[/tex]

    2. Relevant equations

    For all of these, the limit of [tex]\Delta[/tex]X approaches 0.

    3. The attempt at a solution

    The problem is, I can't attempt it because it's different from other ones. I can do simpler things like y(x) = 9x^2 no problem. These are different though.
     
  2. jcsd
  3. Apr 9, 2009 #2
    I'm guessing that you are needing to find y'(x) in each case using the definition of the derivative, which is the limit of the difference quotient.

    I'll help you with the first one, and maybe that will get you started. You need to know this special limit to do the first one:
    [tex]\lim_{x\to0} \frac{1 - cos x}{x} = 0[/tex]
    You also need the trig identity:
    [tex]\cos(u+v) = \cos u\cos v - \sin u\sin v[/itex]

    Then the derivative is:
    [tex] y'(x) = \lim_{h\to 0} \frac{\cos(x+h) - \cos x}{x} [/itex]
    Now apply the above information to compute y'(x).
     
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