Finding the derivative of a polynomial is relatively straight forward, but what about other functions? This activity explores a function with a special property; “the derivative of the function is the same as the original function”! The problem is initially explored using a dynamic line where changes to the line immediately affect the derivative of the line. Manipulating this line provides for instant feedback helping students determine the likely shape of such a function. Calculations involvin
Establish and use the formula d(e^x)/dx)=e^x [ ACMMM100 ]
Gradient, derivative, polynomial
About the Lesson
Students begin by manipulating a flexible line focusing on the outcome of producing a gradient line that matches the original. The flexible line consists of a series of line segments; the gradient of each line segment is instantly computed and plotted. As the basic curve is established it is assumed to be a polynomial, the degree of which is unknown. The equation for the polynomial is generated by successive differentiation and substitution until a pattern is established. The level of calculus used does not go beyond the derivative of a basic polynomial however it does introduce some strategies for determining derivatives of other functions.