# Plotting Derivative Graphs: How to Find f'(x)

• MiniTank
In summary: sorry about that, in summary, the derivative of a piecewise function is discontinuous at certain points, which means its not a true derivative.
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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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.

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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?

We can't 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 won't 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.

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 )

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.

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

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

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can you progress in order of x, your describing separate pieces, I don't think what you just described is understandable.

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

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

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

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

horizontal line?

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.

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?

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

MiniTank said:
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?

You started with a piecewise function.

What's wrong with a piecewise derivative?

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

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.

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

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

## 1. What is a derivative graph?

A derivative graph is a graph that shows the rate of change of a function at every point. It is also known as the slope or gradient of a function, and is represented by the function f'(x).

## 2. Why is it important to plot derivative graphs?

Plotting derivative graphs helps us understand the behavior of a function and its rate of change. It also helps us identify critical points and points of inflection, which are important in finding the maximum and minimum values of a function.

## 3. What is the process for plotting derivative graphs?

The process for plotting derivative graphs involves finding the derivative of the given function, f'(x), and then evaluating it at different points to get the slope of the function at those points. These points and their corresponding slopes can then be plotted on a graph to get the derivative graph.

## 4. How do you find the derivative of a function?

The derivative of a function can be found by using the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules allow us to find the derivative of a function by manipulating its algebraic form.

## 5. What are some common mistakes to avoid when plotting derivative graphs?

Some common mistakes to avoid when plotting derivative graphs include forgetting to evaluate the derivative at each point, using the wrong rule of differentiation, and not considering the domain of the function. It is also important to check for any discontinuities or asymptotes in the function, as they can affect the derivative graph.

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