# Intuitive way to find the graph of a derivative?

I have a homework problem that isn't really explained in my book (Stewart strikes again), I am suppose to match graphs of functions to a graph of its derivative. Is there any easy any logical way to do this just by looking at the graph?

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Stop expecting to be given only problems that you've been shown how to do. That's like learning to drive a car only to those places you've been shown how to drive to. If all the concepts have been explained that you need in order to figure it out, that should be enough. – Michael Hardy Sep 26 '11 at 2:04
That you would get math be reading the proof of a theorem was never suggested. But no one ever gets math until they go beyond simply doing what they've been shown how to do. – Michael Hardy Sep 26 '11 at 2:23
Of course, I am just attempting to study for a test that I am feeling like I will most likely fail. I have had tutors and office hours, plus doing all the assigned homework and a little extra. Looking up videos on kahn academy and then I studied pretty much all weekend. But I guess I am just expecting a free ride through this class because I am stupid and lazy. – user138246 Sep 26 '11 at 2:25
No one has suggested that you're expecting a free ride because you're stupid or lazy, and you shouldn't attribute that view to me. But it remains a fact: No one ever gets math until they go beyond simply doing what they've been shown how to do. And learning math consists mostly of that. – Michael Hardy Sep 26 '11 at 2:27
I attempt to do that, but mostly I just struggle through the homework. – user138246 Sep 26 '11 at 2:28

The key to an exercise like this is to remember that the $\textit{sign}$ of the derivative tells you about the $\textit{rate of change}$ of your function. Remember that saying that the derivative is positive at a point means exactly that it is increasing near that point, that a zero derivative means it is locally neither increasing nor decreasing, and that a negative derivative means (you guessed it) that it is locally decreasing. So a good first step in a problem like this is to identify the regions on which your function is increasing, where the derivative is zero (which could mean a local minimum, a local maximum, or neither), and where it is decreasing, and to match this up with the signs of the derivative.

If that's not enough, you'll have to start looking at second derivatives. These tell you about the rate of change of your derivative: is your function increasing at an increasing rate (think about $x^2$ for $x > 0$) or is it increasing at a decreasing rate (like $-x^2$ for $x < 0$)? Again, don't worry too much about quantitive information; it's usually sufficient in a problem like this to think about heuristics like the sign of your derivative, and how "fast" your function is growing.

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Thanks, that makes sense to me. I will be sure just to take my time with these if they come up on a test. – user138246 Sep 26 '11 at 2:16
"saying that the derivative is positive at a point means exactly that <b>it</b> is increasing near that point" (emphasis added). In fact that the derivative is positive at a point means that the ORIGINAL FUNCTION is increasing at that point. Not that "it" (the derivative?) is increasing at that point. I've seen people get confused about this, so it's worth mentioning. – Michael Hardy Sep 26 '11 at 2:46

The derivative at a point is the slope of the tangent to the graph of $y=f(x)$ at that point. Imagine a point moving along the original graph, and the tangent to the graph at that point. As the point moves along the graph, the slope of the tangent changes. That tells you how the derivative is changing.

When the slope of the tangent is large and positive, the derivative is large and positive. If the tangent is going towards the horizontal, the derivative is going towards $0$. If the tangent has negative slope, the derivative has negative slope. And so on.

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There are a few general rules you can use that will usually narrow the graphs down to one. First, keep in mind that any maximum or minimum of the graph of your function will be at a point where the derivative takes the value 0 (because the tangent there is horizontal.) And in general, the derivative will be positive when the function is increasing and negative when it is decreasing.

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Remember that when you take the derivative of a function, you will end up with a polynomial of one less degree (e.g. derivative of x^2 is x), so a quick check is to make sure that the graph follows this.

This doesn't provide a great deal of information, but it can be used to quickly eliminate possibilities. The other tricks provided should carry you through.

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