Explain the differential function clearly to the math tutor

What you need:

• Paper and pencil for sketches
• calculator

This is how you explain the differential function in calculus

1. Usually the differential function is introduced via the slope of a tangent. The focus of interest is the question of the slope of a function.
2. Perhaps you will start with a very simple (and well-known) case, namely one Straight lines. In the case of straight lines y = mx + b, the slope is relatively easy to determine, it is the number "m" that is in front of the x. The greater the slope m, the steeper the straight line. If the "m" is negative, the straight line falls. Until then there is usually no mental problem.
3. Now select the normal parabola y = x² as the next example. The function graph should be recorded.
4. It quickly becomes apparent that this function has different slopes in individual points. For example, the slope at x = 0 is actually zero, at x = 2 it is greater than at x = 1. One can try to create tangents that reflect the gradient behavior of the function and (with gradient triangles) determine its gradient - a graphical approximation of the problem.
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5. But how can one approach mathematically and thereby develop the differential function? Here too, before generalization, calculation examples help.


Function - calculation of b

The constant "b" is to be calculated for a function. It can only be ...

6. Stay with the normal parabola and, as an approximation for the slope of the tangent, first place secants on the parabola. For example, if you want to calculate the tangent slope at point P0 (2/4), select P1 (3/9) as the first auxiliary point and calculate the slope of the corresponding secant (slope triangle). This slope is of course not a good value, so you have to move the point closer, for example P2 (2.5 / 6.25). Again calculate the slope of the secant.
7. Create a table in which you enter the points P1, P2 etc. Enter the value for the slope behind it. Keep halving the distance to P0. After three to four steps at the latest, the student will notice that there is a limit value for the calculated slopes (namely 4), which then corresponds to the tangent slope in P0.
8. Of course, this calculation and table procedure could be repeated over and over again for every point in the parabola and for every function... but that takes time and patience. So a general calculation basis (and even better: a formula) would be just the right thing to solve the problem once and for all.
9. And you are already at the generalization, namely the differential function, which is nothing more than a Consideration of the limit value for the secant slopes if the sample point is getting closer and closer to the point for which the Want to calculate the slope.
10. And this differential function can be set up for any function, not just for Parabolas. In this way, when considering the limit values, one finally arrives at the derivation rules, for example for power functions.