Derivatives and Constants
Introduction
So far, we know just a few derivatives, and it is time to expand our knowledge by considering the effects of constants. In this notebook, we will assume only the very basic derivatives of trig functions: 
, 
.
We'll begin by reminding ourselves of the graphs of sine and its derivative, cosine. Here, and in the examples below, we'll use red for the original function and blue for the derivative. Evaluate the cell below.
What happens if we add a constant to the function? What happens to the graph? What do you think will happen to the derivative? Evaluate the cell below to define .
Evaluate the next cell to see the graphs of f and f '.
Compare this to the graph of the derivative of 
above. Was this what you expected? What is the equation of 
? State the result in general terms and explain why this should make sense.
Clear the definition of f.
Go back to the definition of f above and change the constant. Evaluate the cells again to see the effects. (Be sure you are convinced about the effect on the derivative.)
In this section, we consider the horizontal translation of the function and how this affects the derivative. We will follow the basic format above.
Plot the function and its derivative.
What happened? What is the equation of g'[x]? Clear g below. Go above again to redefine your function g by changing the constant. Check the derivative graphs.
State what happened. Explain why your result should seem reasonable.
What happens to the graph of a function h when it is multiplied by a constant? What do you think will happen to the derivative? First define h below.
Then plot the graphs of h and h'.
What function is h'[x]? Is this what you expected? Does this make sense? Clear h below. Above, redefine h with another constant and check to see that the effect of the constant is the same each time.
State the effect of a on the derivative of h. Explain why this is logical.
In this situation, what is the effect of a on the basic graph of 
? How do you think it will affect the derivative? Will it affect its period? Will it affect the amplitude of the derivative graph? Will there be some other change?
First define k below.
Then plot k and its derivative.
What is the equation of the function k'[x]?
Then clear k below.
Now redefine k above to be each of the functions below. Graph each derivative and find its equation. In each case, write the function and its derivative.
Now state the general rule for the derivative of k[a x]. Explain why this is logical.
