# Vertical Tangent

The vertical tangent is explored graphically.

Function f given by

*f(x) = x*

^{ 1 / 3}and its first derivative are explored simultaneously in order to gain deep the concept of vertical tangent in calculus.

## Interactive Tutorial

1 - Three graphs are displayed: in blue color the graph of function*f*. The tangent line (in red) to the graph of

*f*and in green color the graph of the first derivative

*f '*which is drawn as the position of the tangent line is changed using the red button slider along the green line.

2 - Use the red button to move the tangent line close to the point whose *x* coordinate is equal to 0.
What happens to the slope of the tangent line? The tangent line is (or almost) vertical.
Calculate the first derivative of *f(x) = x ^{ 1 / 3}*. Is f '(0) is defined?

Use the last result to explain what happens to the slope of the tangent line at

*x = 0*and also to find out if the first derivative has any vertical asymptote at

*x = 0*.

## More References and Links

Derivatives of Sine (*sin x*) Functions . The derivative of sine functions are explored interactively.

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.

First and Second Derivatives Theorems .

Derivative of tan(x) . The derivative of tan (x) is explored interactively to understand the behavior of the tangent line close to a vertical asymptote.