How To Graph A Derivative Of A Function

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  Is the tangent line the derivative of the part at the sure point?

  

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    Almost...the Gradient of the tangent line is the derivative of the function at some certain point.

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  sin(ten) is a abiding part but its slope cos(x) isn't a directly line. why?

  

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    no sinx is non a constant function. Exercise you mean that it is a role that cycles? Because it does do that. Merely it isn't abiding. A function that has a slope of a straight line is a parabola. Merely sinx isn't a parabola. Information technology's slope increases and so decreases so increases and then decreases. That's why the slope of sinx is cosx.

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  Is there a case in which nosotros accept a curve in the derivative, or are there only lines?

  

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    A linear function is a function that has degree ane (as in the highest ability of the independent variable is 1). If the derivative (which lowers the caste of the starting role past 1) ends up with one or lower as the degree, it is linear. If the derivative gives you a caste higher than 1, it is a curve.

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  I recall, at the "first point" of f(x), if I can say it like that, the derivative also shouldn't be defined, because we just cannot make a tangent line over in that location.

  

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    I assume y'all hateful the point x = 0? Well-nigh texts on uncomplicated calculus would not define the derivative at such a point. Such texts unremarkably only ascertain the derivative of a role at an interior bespeak of said function'due south domain. Informally speaking, a indicate is an interior point if we may "approach" it from both sides, while still being in the domain of the function.

    More formally: let A be a nonempty set of real numbers, and suppose a ∈ A. We phone call a an interior point of A if and only if there exists some open interval, containing a, which is entirely contained in A. If a is such a point, we may approach it from both sides from inside A.

    Then, for most elementary purposes, you lot are correct, the derivative is not defined at that place. However, more advanced texts may ascertain what is called a one-sided derivative (see, due east.g., http://en.wikipedia.org/wiki/Left_and_right_derivative for a brief overview). Other texts, such as Analysis I (Tao, 2nd ed., 2009, p. 250) does not make the distinction between the "ordinary" derivative and the one-sided derivative (this is an honours level text on introductory real analysis - not well suited for a beginner).

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  whats the difference between the shaded and not-shaded circles ?

  

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    Shaded circles hateful that the role reaches that loin and information technology'due south defined there, while open circles mean that the function approaches infinitely close to that point but never reach it, then the role is undefined on the open point.

    This is related to the deviation betwixt the minus-than ( < ) and greater-than (>) and the minus-or-equal-than (<=), and the greater-or-equal-than (>=).

    For example, if a function is defined in the domain 2 < 10 <= 5 then the value on the two would be represented by an open circle (since the function approaches it but never actually reaches it), while the 5 would be a filled circumvolve, since the function is defined at that place.

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  Around

  4:24

  , Sal draws a open up dot to offset the blue line - shouldn't it be closed because nosotros know it'southward zero?

  

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  At

  two:00

  , should the graph of the derivative resemble something like the graph of tangent? i.e. this form http://world wide web.purplemath.com/modules/trig/graphs22.gif
  The slope of the semicircle is extremely positive at offset and gradually decreases to zero, and does not decrease at a constant value equally is shown at

  2:00

  

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    If the original graph is of a parabola, rather than a circle, then the graph of the derivative is a directly line, since d/dx[ax² + bx + c] = 2ax + b
    If the original graph is a circumvolve, then the graph of the derivative volition exist like (but opposite) to the purple math epitome you linked to. The graph will wait similar this: https://world wide web.desmos.com/computer/uoe1bollo2
    There will be vertical asymptotes at the left and right edges of the circle. As we move along x from x=0, the derivative will exist very positive, gradually reducing to zip at x=<circle radius>, (where the slope is parallel to the x axis), and and so the graph of the derivative will go more and more than negative.

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  At

  iii:55

  , the ruddy colored line is increasing, only is in f(x)<0. So, I become that it volition accept a constant f'(x) slope, but shouldn't it be in f'(ten)<0.

  

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    Yep,
    f(x) is negative, simply f ' (x) (or F Prime) will be positive, since it is essentially the slope of the line and the slope at that signal is positive.

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  Why is the derivative of second and third office a horizontal line? I mean I don't sympathize how to correspond derivative of a role graphically.

  

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    I'm not quite certain that I empathize your question. Are you request what the second and 3rd derivatives look similar (i.due east. d^2/dx^2, d^3/dx^3)?
    In that instance, await at your horizontal line. What is its slope? Cypher, right? That is the outset derivative. The beginning derivative of a horizontal line is 0 everywhere. What is the gradient of that role (the second derivative)? It is a flat line, y=0, then the slope must still exist 0 which means that the second derivative is everywhere goose egg.
    

    Continue this train of thought, and it is clear that whatsoever derivative (first, second, third...17th...37th) of a horizontal line must always be 0.

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  Why the derivative of a sharp plow is not possible?

  

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    The derivative is basically a tangent line. Recollect the limit definition of a tangent line. As the 2 points making a secant line get closer to each other, they approach the tangent line. With a sharp turn like a cusp, there is no point that the secant line approaches. I hope that makes sense!

Source: https://www.khanacademy.org/math/old-ap-calculus-ab/ab-derivatives-analyze-functions/ab-connecting-func-and-derivatives/v/intuitively-drawing-the-derivative-of-a-function

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