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Thread: Mind blown - probably missing algebra concept - cannot simplify derivative

  1. #1
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    Mind blown - probably missing algebra concept - cannot simplify derivative

    hi all!

    sorry may have been at this too long but ive been blanking on this one for a while now, been coming at it different ways and I cannot work it down!

    f(x) = x^2/sqrt(x^2-1)

    so what i did for f'(x)

    f(x) = x^2 * 1/sqrt(x^2-1)

    product and chain rule to:

    f'(x) = 2x/sqrt(x^2-1) - x^3/(x^2-1)^(3/2)

    I cannot seem for the life of me to algebra this little rascal down a single term and I feel like I should be able to as solving in this state for the critical points is proving....lets use challenging.

    Would really appreciate a point in the right direction!

  2. #2
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    Quote Originally Posted by frank789 View Post
    hi all!

    sorry may have been at this too long but ive been blanking on this one for a while now, been coming at it different ways and I cannot work it down!

    f(x) = x^2/sqrt(x^2-1)

    so what i did for f'(x)

    f(x) = x^2 * 1/sqrt(x^2-1)

    product and chain rule to:

    f'(x) = 2x/sqrt(x^2-1) - x^3/(x^2-1)^(3/2)

    I cannot seem for the life of me to algebra this little rascal down a single term and I feel like I should be able to as solving in this state for the critical points is proving....lets use challenging.

    Would really appreciate a point in the right direction!
    Much easier to apply quotient rule.

    [tex]\frac{d}{dx} \left[\dfrac{g(x)}{h(x}\right] = \ \dfrac{g'(x)*h(x) - g(x)*h'(x)}{[h(x)]^2}[/tex]

    g(x) = x^2 → g'(x) = 2x &

    h(x) = (x^2 - 1)^(1/2) → h'(x) = x * (x^2 - 1)^(-1/2)

    Now continue.....
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
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    ok I went through product rule

    i got

    2x(x^2-1)^(1/2) - (x^3)(x^2-1)^(-1/2)
    ----------------------------------------------
    x^2 - 1

    then I dived each term by the x^2 - 1 to get

    (2x)(x^2 - 1)^(-1/2) - (x^3)(x^2 - 1)^(-3/2)

    which if im not mistaken is the same answer I reached before. as im typing this the first few sips of coffee #2 seem to have done the trick

    factor out the largest power of (x^2 - 1), which is (-3/2), along with an x to give

    x(x^2 - 1)^(-3/2)(2(x^2 - 1) -x^2)

    i expanded inside my binomial and collected like terms to x(x^2 - 1)^(-3/2) * (x^2 - 2)

    and I guess since I bothered to type this out I may as well verify I would solve x(x^2 - 2) = 0 for the critical points and get the other critical points from (x^2 - 1) ^ (-3/2) = 0 due to domain restrictions.

    thanks again for everything and sorry for posting something I ended up figuring out (im pretty sure anyway), but its amazing what someone verifying your right up until that point does for confidence to finish.

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