An Intuitive Approach to the Derivative
- Author:
- Ken Schwartz
- Topic:
- Derivative, Difference and Slope
Let's look at the Derivative simply for what it is - a function that gives us the slope of another function.
The basic idea of the derivative is actually pretty simple - it's the function that gives the slope, rather than the height (-value), of a function at each value of . One way to think about this is that if the point on the graph of at is , -value of at , then the point on the graph of at would be , slope of at .
As you work with this app, check out Khan Academy's Derivative Intuition Module at https://www.khanacademy.org/math/differential-calculus/taking-derivatives/visualizing-derivatives-tutorial/v/derivative-intuition-module. This app is based on the same concept, but it allows YOU to move the dots, and also lets you try many different functions.
To start with, be sure the "Show Derivative" checkbox is cleared (unchecked). The graph of is shown in green. Different functions can be investigated by selecting one from the drop-down list at the top right. You can even enter your own "user" function in the "f(x) =" box; it will then appear at the bottom of the drop-down list. Now the task is to slide the red dots up or down so that the red line segment at that value of is tangent to (matches the slope of) at that point. The -value of the red dot is the slope of the corresponding red tangent line segment. Since the derivative function gives us the slope of the original function , the red dot should therefore lie on the graph of . Adjust all the dots so that all the segments are tangent to the green graph.
Now check the "Show Derivative" box. The graph of , the derivative of , will appear in blue. If you were successful in adjusting the tangent line slopes, your red dots should lie on or very close to the graph of .
A special function is . This is the second-to-last function in the drop-down box. You can drag the graph down and/or use the Zoom buttons to see more points. Make note of the result you get for this function - you'll see this many more times in this course!