How to Use a Derivative Plotter to Boost Calculus Learning
A derivative plotter typically plots a derivative function in blue and also plots the slope of the function on the graph shown in red (by computing the difference between each point in the original function, so it is not familiar with the formula for the derivative). In addition, you also have the option to plot another function in green below the computed slope. If the lines coincide with each other there are fair chances that you have found the derivative!
[Image will be uploaded soon]
Derivative Grapher Generator
Shown below is the graph of f(x).
The point marked "Slope of The Tangent Line contains a value ‘y’ which is the slope of the line tangent to the point "Drag Me"
Now, you can plot the graph of the derivative of this function by
1. Clicking on the "Begin Graphing the Derivative" button, then
2. Moving the point "Drag Me"
The graph of the f '(x) will be tracked down onto the screen.
When you click on "Stop Graphing The Derivative" the graph will withdraw.
[Image will be uploaded soon]
Change the equation of f(x) in the top left corner for the purpose of changing the function.
For each of the following functions, set a comparison of the graphs of the derivative of the functions to the common functions that you are known of. Can you determine the equation of the derivative functions?
Example:
Show that the function f(x) = x3– 2x² + 2x, x ∈ Q is increasing on Q.
Solution:
F(x) = x3 – 2x² + 2x
Upon differentiating both the sides, we obtain,
f'(x) = 3x² – 4x + 2 > 0 for each value of x
Hence, f is increasing on Q.
Derivative Visualization
Visualization of Derivative shows 4 different functions, which we can switch by pushing "Switch Function". It also consists of two points, F and M, and you can drag point M with the help of the slider. You can also display the tangent and chords using the derivative visualization.
Now let's play around until you understand the buttons
[Image will be uploaded soon]
Step 1: We want to identify the slope of the tangent line, which travels through point F on the function. Push the function "Show Tangent at F".
Step 2: We can now easily identify the slope between two points if we have their coordinates. Thus, if we have a point M, we can construct a chord across the two points, and identify the slope. Push "Show Chord Across F and M".
Objective: Now, we want to measure the slope of the tangent line at F, with the help of the chord through F and M.
Interactive Derivative Graph
Interactive Graph displaying Differentiation of a Polynomial Function.
In the following interactive derivative graph, you can explore how the slope of a curve changes as the changes the variable x.
Things to Do For Derivative Graph Calculator
Below we have taken the Derivatives of Polynomials.
In the left panel, you will notice the graph of the function of interest, and a triangle having a base 1 unit, stating the slope of the tangent. In the right panel is the graph of the 1st derivative (the dotted curve).
Take the help of the slider at the bottom in order to change the x-value. You can move the slider right or left (keeping the cursor within the light gray area) or you can also animate the points by holding down the "+" or "−" buttons on either side of the slider.
[Image will be uploaded soon]
Select another of the 2 examples in the pull-down menu.
The height of the right triangle implies the slope. It consists of a base of 1 unit.
Derivatives of 3 Functions
Below are the derivatives of the 3 functions:
1. Quadratic (parabola) y = x² - 10x -1
Derivative: dy/dx = 2x − 10
2. Cubic, y = 0.015x3 −0.25x² + 0.49x + 0.47.
Derivative: dy/dx = 0.045x² − 0.5x + 0.49
3. Quartic y = x4 − 1.5x3 − 6x² + 3.5x + 3.
Derivative: dy/dx = 4x3 − 4.5x² − 12x + 3.5.
FAQs on Derivative Plotter: Graphing and Understanding Derivatives
1. What is a derivative plotter and what does it show?
A derivative plotter is a tool that generates a visual graph of a function's derivative. For a given function f(x), it plots its derivative f'(x). Each point on this new graph represents the instantaneous rate of change or the slope of the tangent line of the original function f(x) at the corresponding x-value. It is used to visually understand how a function's rate of change behaves.
2. What is the relationship between the graph of a function and its derivative plot?
The relationship is key to understanding calculus concepts visually. The derivative plot, f'(x), directly describes the characteristics of the original function's graph, f(x). Key examples include:
- The y-value on the derivative plot equals the slope of the original function at that x-point.
- A peak or valley (local maximum or minimum) on the original function's graph corresponds to an x-intercept (where the graph crosses the x-axis) on the derivative plot.
- An inflection point on the original graph corresponds to a local maximum or minimum on the derivative plot.
3. How can you use a derivative plot to determine if a function is increasing or decreasing?
A derivative plot provides a simple visual test for a function's behavior as per the CBSE syllabus on Application of Derivatives. The rules are:
- If the derivative plot is above the x-axis (meaning f'(x) > 0), the original function f(x) is increasing in that interval.
- If the derivative plot is below the x-axis (meaning f'(x) < 0), the original function f(x) is decreasing in that interval.
4. What is the significance of the points where the derivative plot crosses the x-axis?
A point where the derivative plot crosses the x-axis signifies that f'(x) = 0. This corresponds to a stationary point on the original function's graph. These points are critical for finding local maxima and minima. If the plot crosses from positive to negative, it indicates a local maximum. If it crosses from negative to positive, it indicates a local minimum.
5. What is the importance of using a derivative plotter for studying calculus?
Using a derivative plotter is important because it transforms abstract mathematical rules into intuitive visual information. It helps students to:
- Visualize Concepts: Connect the algebraic concept of a derivative to the geometric concept of slope.
- Analyse Functions: Quickly identify intervals of increase/decrease and locate potential maxima and minima without complex calculations.
- Verify Answers: Check the results of manual differentiation and function analysis, which is a useful skill for exam preparation.
6. As an example, what does the derivative plot of a simple parabola like f(x) = x² look like?
The derivative of f(x) = x² is f'(x) = 2x. Therefore, the derivative plot is a straight line passing through the origin with a slope of 2. This visually confirms the behavior of the parabola:
- When x < 0, the parabola's slope is negative, and the derivative plot y=2x is below the x-axis.
- At x = 0, the parabola has its minimum point (slope is zero), and the derivative plot crosses the x-axis.
- When x > 0, the parabola's slope is positive, and the derivative plot is above the x-axis.
7. Can you also plot a second derivative, and what does it reveal about the original function?
Yes, you can plot the second derivative, f''(x), which is the derivative of the first derivative. According to the CBSE Class 12 syllabus for 2025-26, the second derivative provides crucial information about the concavity of the original function f(x).
- If the second derivative plot is positive (f''(x) > 0), the original function is concave up (shaped like a cup).
- If the second derivative plot is negative (f''(x) < 0), the original function is concave down (shaped like a cap).
- Points where f''(x) = 0 and changes sign indicate points of inflection on the original graph.
