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so when I graph a derivative I know how to use the negative/positive rule according to the slope of the tangent line. But what im really struggling with is the concavity of the derivative graph lines. How do I know which way the derivative graph lines are supposed to face?

As an extended question can you also give me a brief introduction on how to graph the derivative of asymptotes. What must I do different compared to the other graphs.

Thanks so much!

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    hmm..just a suggestion, read the numbers and try to forget the graphs. it helps to understand, but also somehow hinders your way to mathematical abstraction. Regarding your question, concave/convex can be determined by second derivative. Or, if you want to read from graph: take a point you want to investigate, if in the neighborhood of that point the function curve is above the derivative graph line, then it is convex, otherwise it is concave. This approach only works for single variable function. If the derivative line is vertical, then no derivative exists, as you may know.2011-09-28
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    If I understand your question, you want to determine where $y=f'(x)$ is facing up, down. Usually, this issue does not arise in calculus problems, where the emphasis is on the relation between the shape of $y=f(x)$ and $y=f'(x)$. If you are given a rough sketch of $y=f(x)$, you will not succeed in doing an accurate sketch of $y=f'(x)$, including concavity of $y=f'(x)$. If you have a *formula* for $f'(x)$, that is another matter, the concavity of $f'(x)$ is determined by the sign of $f'''(x)$.2011-09-28
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    I find it hard to make sense of this question. What does "graph a derivative" mean? Are you drawing the graph of the function $f$ with the help of its derivative $f'$, or are you really saying that you want to draw the graph of the function $f'$?2011-09-28
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    I mean sketching the graph of the derivative on the basis of another functions graph. In between critical points, how do I know which way the concavity is facing? Andre are you suggesting that just on the basis of the graph I should just connect the points with rough curves and disregard the concavity?2011-09-28
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    @John: Oh, I see. So you want to know whether you should draw the derivative's graph $y=f'(x)$ convex or concave? That would be determined by whether the *third* derivative $f'''$ is positive or negative, and (as André says) that seems quite hard to decide just by looking at the graph of $f$.2011-09-28
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    Like they show me a derived graph without the equation. Then they ask me to sketch the original graph giving me one point through which the line has to pass to. I have know established like a small set of rules for myself, but is a guide on the net? For example when the derived graph is a "hill" on the positive y plane then the original graph must first have a slope "z" - then increase to a steeper slope "z+2" and decrease back to slope z - while the actual slop must remain positive. Sorry if this is hard to visualize. But i hope you understand what I mean. Could you please point me in the2011-09-28
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    right direction2011-09-28

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