The first derivative of tan (x) ,as defined in calculus, is explored graphically and interactively.
Function f(x) = tan(x) and its first derivative are explored graphically in order to gain deep understanding of the concept and the behaviour of the derivative near vertical asymptotes.
Interactive Tutorial1 - Click on the button "click here to start" and maximize the window obtained. Three graphs are displayed in the open interval (-Pi/2 , Pi/2): the graph of f in blue. In red color the tangent line to the graph of f and in black color the graph of the first derivative f ' which is drawn as the position of the tangent line is changed using the bottom slider ("Change Tangent Position").
2 - Use the bottom slider to change the position of the tangent slowly from left to right. What is the sign of the first derivative (in black)? What is the sign of the slope? Use the first derivative of tan(x) to explain your findings.
3 - tan (x) has vertical asymptotes at x = -Pi/2 and at x = Pi/2. Use the bottom slider to move the tangent line from left to right. Observe what happens to the slope of the tangent line as you approach the vertical asymptote of the graph of tan (x) (in blue). Observe also what happens to the first derivative. Use the first derivative of tan (x) to explain what you observed.
More on derivatives:
Derivatives of Quadratic Functions. The derivative of quadratic functions are explored graphically and interactively.
Derivatives of Polynomial Functions. The derivative of third order polynomial functions are explored interactively and graphically.
Derivatives of Sine (sin x) Functions. The derivative of sine functions are explored interactively.
Vertical Tangent. The derivative of f(x) = x 1 / 3 is explored interactively to understand the concept of vertical tangent.