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Math Help - Relationship between differentiation of inverses

  1. #1
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    Relationship between differentiation of inverses



    (image taken from Karl's Calculus (Karl's Calculus Tutor - 4.3 Derivatives: More Rules to Live By)

    There are 2 wavy lines both starting from origin (lets call the wavy line that starts at the upper part TOP and the lower line BOTTOM) following the red line y=x. One is the inverse of the other. ( Remember that inverses are reflected off the y=x line).

    At the same time, there is a sinosoidal curve which is supposed to be the derivative (the rate) of one of the wavy line TOP.
    Mr Karl has plotted the derivative of the other wavy line (the BOTTOM) as a series of spikes, reflecting the fact that ({1 \over 8})^{-1} = 8.

    But is this characteristic of all derivatives inverses? Wont the derivative be more correct if its just another sinusoidal curve (phased shifted by half a cycle) instead of the spikes?

    Please comment.
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by chopet View Post


    (image taken from Karl's Calculus (Karl's Calculus Tutor - 4.3 Derivatives: More Rules to Live By)

    There are 2 wavy lines both starting from origin (lets call the wavy line that starts at the upper part TOP and the lower line BOTTOM) following the red line y=x. One is the inverse of the other. ( Remember that inverses are reflected off the y=x line).

    At the same time, there is a sinosoidal curve which is supposed to be the derivative (the rate) of one of the wavy line TOP.
    Mr Karl has plotted the derivative of the other wavy line (the BOTTOM) as a series of spikes, reflecting the fact that ({1 \over 8})^{-1} = 8.

    But is this characteristic of all derivatives inverses? Wont the derivative be more correct if its just another sinusoidal curve (phased shifted by half a cycle) instead of the spikes?

    Please comment.
    partly right partly wrong..

    what is the definition of an inverse? in functions, two functions are inverses (we disregard the property of one-to-one: i.e. we restrict the domain so that the function is one-to-one, and we do this every period) if you compose them, then it gives you the indentity function..

    let us consider a continuous function on a point by point basis.. what if the value of the function at x=a is 0, then what is the inverse of zero?

    as in the example, if the derivatives are another function, you can observe that there are points on the blue sinosoidal curve (the derivative one) which approaches y=0, hence its inverse approaches infinity at these points..

    besides, observe the original function and its inverse.. if you look at the points of intersection, the blue one is almost horizontal while the green one is almost vertical.. and what is the slope of the tangent line (or the derivative) if these were the situation? (this fact coincides with the previous paragraph..)

    i hope you got what i mean..
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  3. #3
    GAMMA Mathematics
    colby2152's Avatar
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    Your logic is great here, but it is misleading. Follow the curves of the function and its derivative, and then do the same for the inverse of the function and its corresponding derivative. The resulting graphs of the derivatives will then make sense to you.
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