The tangent problem

The gradient of the tangent

The derivative of a function at a point


One of the main difficulties students encounter when beginning to study the derivative of a function is understanding it in geometrical terms. Whereas calculating derivatives is usually fairly straightforward and can even be quite appealing, applying the geometric interpretation of a derivative at a point becomes a more complex problem. This is often due to not having a clear understanding of the concept, rather than the problem itself being too complex. 

The activities which follow are designed to familiarise the student with the concepts of the secant and tangent to a curve, to observe how the former approaches the latter and to understand the limit as a process which can be observed and checked.

  • To identify the problem of drawing the tangent to a curve at a point
  • To identify the tangent as a limit of the secants.
  • To determine that the gradient of the tangent is the limit of the gradients of the secants.
  • To obtain a graphical representation of the derivative of a function at a point.
  • To use the derivative in order to determine the equation of a tangent to a curve at a point.

  Juan Madrigal Muga
Spanish Ministry of Education and Education. Year 2001

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