# 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)*

## 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 annd 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*is explored interactively to understand the concept of vertical tangent.

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