# Derivative graphs

1. Apr 25, 2005

### MiniTank

how exactly do you go from a graph of a function to plotting the graph of the derivative of its function?

ex: y=f(x) .. this is just the general shape with the intercepts(check the attatchment)

im not sure but when going from the original function to its derivative, does the function lose a turning point, making this a straight line?

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Last edited: Apr 25, 2005
2. Apr 25, 2005

### Data

Well, first let's assume the function that you're given is differentiable. You can start with maxima and minima: wherever the function has a maximum or minimum, the derivative is zero. When the function is getting bigger from left to right, the derivative is positive. When the function is getting smaller from left to right, the derivative is negative.

The magnitude of the derivative depends on how fast the function is increasing or decreasing: if it's increasing very fast, the derivative is very big and positive. If it's increasing more slowly, the derivative is small and positive. If it's decreasing slowly, the derivative is small and negative. If it's decreasing quickly, the derivative is large and negative.

As well, concavity changes can help. A change in concavity of the function represents a maximum or minimum of the derivative. If the function changes from concave up to concave down at a point on the x axis, then the derivative will have a maximum there. If the concavity change is the opposite, (concave down -> concave up), you get a minimum in the derivative.

Last edited: Apr 25, 2005
3. Apr 25, 2005

### MiniTank

ok that means that since that straight line, being the maximum, is from x=-5 to x=2(not in attatchment), then on the derivative graph, between those points i have a straight line where y=o?, now from the left to right, the line sloping up, you can see it gets larger so what is the y-value for these x-values.. the slope? same for the other line sloping down?

4. Apr 25, 2005

### whozum

We cant see the graph yet. If its a line of form y = mx+b, then the derivative is m (the slope). Since its linear, the derivative wont change throughout the whole graph, and your maximum value (assuming m is positive) will be at the highest x value. Your derivative graph is just y = m.

5. Apr 25, 2005

### Data

If you mean that you have a straight, horizontal line (ie. slope is 0), then the derivative is 0 (differentiate it! What's the derivative of a constant?). As I said, if the function is increasing, then its derivative is positive, and if it is decreasing then its derivative is negative. If it is constant, its derivative is 0! (simple enough )

6. Apr 25, 2005

### MiniTank

no... one part is straight.. from left to right .. a line slopes up until y=3, then at y=3, there is a horizontal line from x=-5 to x=2.. then at x=-2 it slopes down and stops at x=5 where as the the line sloping up on the left is continuous to infinite.

7. Apr 25, 2005

### MiniTank

****____________________
***/****************** \
**/********************\
*/**********************\
/

the "*"s are just blanks, if you use spaces , the board autoamtically deletes them, thats the shape.. from right to left ... the left side continues to infinite... you guys understand now?

Last edited: Apr 25, 2005
8. Apr 25, 2005

### whozum

can you progress in order of x, your describing separate pieces, I dont think what you just described is understandable.

9. Apr 25, 2005

### MiniTank

x<=-5;;;;;;;; f(x)= 3x + 18
-5<=x<=2;;; f(x)=3
2<=x<=5;;;; f(x)=-2x +7

10. Apr 25, 2005

### whozum

x<=-5;;;;;;;; f(x)= 3x + 18 df/dx = m = 3
-5<=x<=2;;; f(x)=3 df/dx = m = 0
2<=x<=5;;;; f(x)=-2x +7 df/dx = m = -2

11. Apr 25, 2005

### MiniTank

i understand how you got those values but then what do i do to graph it?

12. Apr 25, 2005

### whozum

What does the graph f(x) = 3 look like?

13. Apr 25, 2005

### MiniTank

horizontal line?

14. Apr 25, 2005

### HallsofIvy

Staff Emeritus
If you mean "what do I do to graph when I have no idea what a derivative IS", then the answer is you DON'T. What everyone has been trying to tell you is that the derivative is the slope of the tangent line. Look at the graph of f(x). If the graph is going up steeply, then the derivative is a large positive number. If it is going down steeply then the derivative is a large [B\]negative[/b] number. If the graph is about "level" then the derivative is close to 0.

15. Apr 25, 2005

### MiniTank

i DO know what a derivative is, its i just don't know how to graph the derivative of a function. like in this situation, what i don't get is, would the graph just have a few horizontal lines?

16. Apr 25, 2005

### whozum

Yep, of values mentioned above during the intervals above.
Graphing derivatives is something you learn very early in calculus..

17. Apr 25, 2005

### Hippo

You started with a piecewise function.

What's wrong with a piecewise derivative?

18. Apr 25, 2005

### MiniTank

its not a piecewise function, its continuous throughout. Just to confirm, piece wise means it is discontinuous at certain points, right? i guess my domain was incorrect. I personally wrote out the domain, I wasn't 100% sure if it was right but the graph should be on continuous function without any gaps

19. Apr 25, 2005

### whozum

no, piecewise means its composed of different functions, not one function running through a domain, but a few functions running through a few different domains.

20. Apr 26, 2005

### LENIN

Yes there are just a few horizontall lines. It`s something like this.

----********************
***********************
***********************
****---------************
***********************
***********---------------