# Derivatives of Sine Functions

The derivatives of sine functions, as defined in calculus, are explored graphically and interactively.

A sine function of the form

*f(x) = a sin (b x)*

and its first derivative are explored graphically and simultaneously in order to gain deep understanding of the properties of the function and its derivative. You may need to review some important calculus theorems in order to be able to fully understand the relationship between the behavior of a given sinusoidal function, its derivatives and the tangent line to the graph of the given function.
## Interactive Tutorial

1 - Three graphs are displayed below: in blue the graph of function f and in red the first derivative f '. The tangent line to the graph of f, in black color, is drawn at the same x-position of the red button (bottom) whose position can be changed by sliding it along the green line.

2 - Slide the red button to change the position of the tangent and note that the tangent line is horizontal (or almost) at the local maximum and minimum of function f (blue). Note also that at these same positions, the first derivative is equal (close) to zero.

3 - Start from a minimum and move the tangent forward up to the next maximum. Over this interval the function increases. What is the sign of *f '* within this interval of increase?

4 - Start from a maximum and move the tangent forward up to the next minimum. Over this interval the function decreases. What is the sign of *f '* within this interval of decrease?

5 - Change the value of b from 1 and to 2, 3, 4 ... where b increases. What happens to the amplitude of the derivative? Find the first derivative of *f(x) = a sin (b x) * and explain analytically what happens when you increase b.

6 - Change the value of a and b and observe and explain the behavior of the function, its derivative and the tangent line.

## More References and Links

First and Second Derivatives Theorems.

Derivatives of Polynomial Functions. The derivative of third order polynomial functions are explored interactively and graphically.

Derivatives of Quadratic Functions. The derivative of quadratic functions are explored graphically and interactively.

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.

Vertical Tangent. The derivative of *f(x) = x *^{ 1 / 3} is explored interactively to understand the concept of vertical tangent.