How can you tell what the derivative of a graph looks like ...

[Jesse_J]

So my math teacher taught us this in class but i kind of forgot.
So, she gives us a picture of a graph (Usually a bunch of random squiggly line stuff) and tells us to find the derivative. I don't have the function so you can't rely on evaluating the function. You just have to look and the graph and know what its derivative graph looks like. Please help!

[Tazate]

if you know anything about the function you can get a pretty good idea. The derivative of a graph that would show position shows velocity, and the derivative of that shows acceleration.

A position graph of something falling in gravity is generally parabolic in shape (or at least one side of a parabola, but the velocity is a constant slope, perhaps reflected upon the point of inversion (when the ball stops going up and starts going down. The acceleration graph is a flat line, as acceleration under gravity is constant.

Generally, if you know the degree of the function (what the highest exponent in it is) you can know what the derivative will look like (one lower degree)

edit:

realized that I forgot to mention what degrees of graphs look like.

The higher the exponent, the more bends the graph will tend to have but that is not always true. x^16 only has one bend, but it has a very steep slope for its derivative. If there is an odd exponent, the graph will go up on one side, down on the other, even it will go up or down on both. if the sign is positive it will go up as it goes to the right, if negative it will go down. tighter curves have higher degrees.

I hope that helps

[Jones]

As I remember it should be that the derivative is the slope of the function aat any given point of time.
So see the function and the way it is traversing and if the slope of the function is from top to bottom then you can say that the derivative is negative and if the slope is positive i.e., if the graph is moving upper side if we move from left to right then we get a positive slope  and if the line is horizantal then we get a zero slope.

Derivative = Slope

[Jones]

Usually I just draw a bunch of tangent lines on the graph in my mind's eye, and determine their signs (positive or negative) and whether their slopes are getting larger or smaller.

Let's say you are presented with the graph of y = x^2. You can clearly see that at the left side of the y-axis, the tangent lines on the curve all have a negative slope. At the coordinates (0 , 0), the slope of the tangent line is 0. Moving from there to the right, the curve has tangent lines with positive slopes. So you know that as general guideline, your derivative graph would move from below the x-axis to above the x-axis, when you're looking at it from left to right. Remember, the derivative graph is just a plot of the slopes of the tangent lines on the normal graph.

Now, estimate the general values of the slopes. The left half of the parabola has tangent lines with slopes gradually getting LARGER, because they're less and less negative moving from left to right. The derivative graph will start from the third quadrant, and move closer to (0, 0). The right half of the parabola will also have slopes that are getting larger, and so will start from (0, 0) and move toward the upper right corner of the first quadrant.

That's just the general shape of the derivative graph. The second derivative will also change how the derivative graph looks; specifically, its concavity (whether the graph is a bowl, upside-down bowl, or just a straight line), but I don't know if you've gotten to that point yet.

Did that make any sense? I feel like I went through it too quickly. D: If you need any more help, search on the internet for something like, "Graphing derivative without function." And here's a Youtube video that looks helpful. (: