Understanding the Derivative-as-a-Function Concept

Many students are not able to conceptualise the derivative of a function as a function in its own right. Some students, for example, equate the derivative of a function with the equation for the line tangent to the graph of the function at a given point. This resource addresses difficulties such as these.

Section 1 of the resource asks students to differentiate a function and explain its meaning. This measures whether 1) students realise that a derivative is a function that allows the calculation of a derivative at any point, and 2) is a function which can be represented graphically. If they do not understand this, Section 2 is designed to help them.

In Section 2, like the ‘Understanding the Meaning of Derivative’ resource, the relationship between the surface area of a balloon and its radius is used as the context to embed the resource within. It is stressed that a function is generated by taking a number of points on the function representing the relationship in question, in order to find out how much the surface area of the balloon increases when the radius of the balloon instantaneously passes through a particular radius value. The section also reinforces what students should have learnt in the ‘Understanding the Meaning of Derivative’ resource, by way of re-iterating the estimation/limiting process associated with finding the slope of a tangent at any particular point on a function.

Students then tabulate their results, and plot a graph of the derivative of the surface area of the balloon with respect to the radius of the balloon against the radius of the balloon, revealing what appears to be a linear relationship. Further questions re-enforce the idea that the derivative of a function is function in its own right. The graph of the original function representing the relationship between the surface area of a balloon and its radius is then shown against the graph of the derivative function, in order to stress the visual nature of the derivative function and its meaning.