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:
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,
Computing the Derivative Algebraically
Step 1. Compute the difference quotient and simplify the answer as much as possible.
Example Let f(x) = 3x2 + 4x. Enter the required expressions using valid technology format.
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 five-day period. Suppose you find that the value of one US Dollar can be well approximated by the function
where t is time in days. (t = 0 represents noon on Monday.)
We would like to compute the exact (instantaneous) rate of change of the value of the US dollar at every time t. Since the instantaneous rate of change of R(t) is given by the derivative, R'(t), that is what we now compute.
One more, this time a little different:
f(x+h) is given by
Now try some of the exercises in Section 3.6 in Applied Calculus or Section 10.6 in Finite Mathematics and Applied Calculus.
Alternatively, press "game version" on the sidebar to go to the game version of this tutorial (it has different examples to try and is a lot of fun!).
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