# 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 -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 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 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!

*slope*, rather than the*height*(**YOU**to move the dots, and also lets you try many different functions. To start with, be sure the "Show Derivative" checkbox is cleared (*un*checked). The graph of*tangent to*(matches the slope of)