3.6 The Derivative: Algebraic Viewpoint
Q So far, all we have been doing is approximating the derivative of a function. Is there a way of computing it exactly?
A Let us start all by recalling what we learned in the previous section: We saw that the derivative of the function f at the point x is the slope of the tangent line through (x, f(x)), or the instantaneous rate of change of f at the point x:
Derivative Function
If f is a function, its derivative function f' is the function whose value f'(x) at x is the derivative of f at x. Its domain is the set of all x at which f is differentiable. Equivalently, f' associates to each x the slope of the tangent to the graph of the function f at x, or the instantaneous rate of change of f at x. Thus,

Derivative = f'(x) = Slope of Tangent at x
Q OK, so how do we compute it exactly?
A We will go through an example to see how. First notice that the derivative is the limit of a certain expression,

\frac{f(x+h)  f(x)}{h}
Computing the Derivative Algebraically
Step 1. Compute the difference quotient and simplify the answer as much as possible. Example Let f(x) = Enter the required expressions using valid technology format. Note: If you have problems anwering the following correctly, you should first try the nongame version of the tutorial. 
Now here is another one for you to try, and we return to a scenario introduced in the tutorial for Section 3.5 (or 10.5 for the combined book):
You are based in Indonesia, and you monitor the value of the US Dollar on the foreign exchange market very closely during a rather active fiveday period. Suppose you find that the value of one US Dollar can be well approximated by the function

The rupiah is the Indonesian currency
where t is time in days. (t = 0 represents noon on Monday.)
One more; but this time a little different:
Q  Let f(x) =  . Then f(x+h) is given by 
Q  \frac{f(x+h)  f(x)}{h}  = 
Now try some of the exercises in Section 3.6 in Applied Calculus or Section 10.6 in Finite Mathematics and Applied Calculus.