Quotient rule for derivatives

Discussion in 'General Math' started by Ryan Stewart, Dec 23, 2004.

  1. Ryan Stewart

    Ryan Stewart Guest

    I'm studying calculus independently, and an exercise in the book I'm using
    is:
    Derive the quotient rule. (Find y' for y = u(x)/v(x).)

    The answer I came up with matches an intermediate step shown in the answers
    section, but not the final answer. I think the book is wrong. My answer:
    u'/v - (uv')/v^2

    Process:
    y = u/v = u(v^-1)
    Product rule:
    f(x) = u, g(x) = v^-1
    f'(x) = u'
    Chain rule to find g'(x):
    Let g(x) = j(k(x)) = v^-1 and k(x) = w = v. Then j(w) = w^-1
    j'(w) = -w^-2, k'(x) = v'
    j'(x) = (-v^-2)v'
    g'(x) = -v'/v^2
    Finally:
    y' = f(x)g'(x) + f'(x)g(x) = -uv'/v^2 + u'/v
    y' = u'/v - uv'/v^2

    The book shows this intermediate step which coincides with my answer:
    y' = u(-1)(v^-2)v' + u'/v

    But the final answer shown is this:
    vu'/v^2 - uv'/v^2

    Can anyone comment on this?
     
    Ryan Stewart, Dec 23, 2004
    #1
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  2. Ryan Stewart

    N. Silver Guest

    Find the lowest common denominator,
    LCD = v^2.
    What you have is the same as
    vu'/v^2 - (uv')/v^2
    = (vu'- uv')/v^2,
    which is the correct answer.
     
    N. Silver, Dec 23, 2004
    #2
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  3. Ryan Stewart

    Ryan Stewart Guest

    Crap. Thanks. That was silly.
     
    Ryan Stewart, Dec 23, 2004
    #3
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