The tables on the preceding page show that the derivative of at the following plotted points in increments of starting left to right x =.
In the following graph, the points are connected to form , .

The following is my investigation of the derivatives of functions in the form. The red dots in the following graphs below represent the derivative of the function at a particular Xvalue.
Concluded from the graphs above, my conjecture for the derivative of the function, is that when, a determines the function amplitude or vertical stretch. Consider also the following graphs:

The following is my investigation of the derivatives of functions in the form. The red dots in the following graphs below represent the derivative of the function at a particular Xvalue.
Concluded from the graphs above, my conjecture for the derivative of the function, is that when, b determines the function frequency or horizontal stretch. Consider also the following graphs:

The following is my investigation of the derivatives of functions in the form. The red dots in the following graphs below represent the derivative of the function at a particular Xvalue. (The gray line is.)
Concluded from the graphs above, my conjecture for the derivative of the function, is that when, c determines the function phase shift or horizontal translation. Consider also the following graphs:

Therefore, based on previous conjectures, the derivative of , where a (amplitude) is 2, b (frequency) is 3, and c (phase shift) is 4, is represented as red dots in the following graph:

Also, based on previous conjectures and the chain rule, the derivative of can be written as. Consider the following graphs where the red line represents,, the gray line represents , and the red points represent the derivative of at an Xvalue.
Because of the chain rule: and,
Thus:
or
=
In conclusion, from my investigation of the Derivative of sine functions, I have discovered the following:

, where the derivative is affected by

a the amplitude

b the frequency

c the phase shift

The derivative of can be written as.
*In this investigation, there are limitations to graphs given as examples. The values substituted only occurred and, therefore, no negative values can be accounted for in the conjectures. Also, the graphs used only fall under and , and, thus, cannot confirm the conjecture outside of these limits.