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 + c)
and its first derivative are explored graphically simultaneously in order to gain deep understanding of the properties of the function and its derivative.
1 - Click on the button "click here to start" and maximize the window obtained. Three graphs are displayed: in blue color the graph of function f. 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 ("Change Tangent Position") to change the position of the tangent line(red). Note that the tangent line is horizontal (or almost) at the local maxima and minima of function f (blue). Note also that at these same positions, the first derivative is equal to zero.
3 - Start from a minimum and move the tangent forward up to the next maximum. 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. What is the sign of f ' within this interval of decrease?
5 - Set parameters a and c to some values and change d. Start from d = 1 and slowly increase it. What happens to the amplitude of the derivative? Find the first derivative of f(x) = a sin (b x + c) and explain analytically what happens when you incrase d.
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.
Derivative of tan(x). The derivative of tan (x) is explored interactively to understand the behaviour 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.