Most Grade 10 syllabus do not yet include the concept and calculation of derivative. At this level, the study of function which started in Year 7 and Year 8 on linear function is simply extended to investigating other function classes such as polynomial function to which linear and quadratics belong, the exponential function and its inverse, rational function etc. There is no mention of derivative. This should not prevent teachers from deriving functions based on the properties of the function students already know.
In my earlier post for example on Application of the Discriminant where the task is to find the equations of the line tangent to the graph of a quadratic function, without using the formula for finding the derivative, a function can be generated between the values of x and the slope of the tangent. The value that can be generated if you take other points is shown below:
You may ask the learners if there exists points on the curve where tangents to those points will be the same. The answer of course is none since for every x there corresponds a unique value of the slope of the tangent (students would need to have several points to convince themselves of this). Then you can ask them to graph the function and to find its equation. It would of course yield y=2x. This is actually the derivative of but you can’t use the term derivative yet. You may however say a derived function from . My point is that doing this task provide learners the opportunity to review discriminant, an opportunity to experience how a new function can be generated from another function by noting varying quantities, and of course have an intuitive notion of the derivative.
You can always extend the exercise by considering other parabola like the one below and other quadratic function.
Is this method generalizable to other polynomial function?
Here’s another way of teaching this using GeoGebra: Teaching the derivative without really trying.