## 3.6 The Derivative: Algebraic Viewpoint

This topic is also in Section 3.6 in Applied Calculus or Section 10.6 in Finite Mathematics and Applied Calculus

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,

 f'(x) = Slope of Tangent at x = limh→0 \frac{f(x+h) - f(x)}{h}

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}
called the difference quotient of the function f. The technique of computing f'(x) is roughly this: Calculate and simplify the difference quotient as much as possible. It is then a simple matter to see what happens to it as h approaches zero.

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 non-game version of the tutorial.

f(x)=
f(x+h)=
f(x+h) - f(x) =  Do not simplify yet.
f(x+h) - f(x) =  Simplified.
 f(x+h) - f(x)h
=

Step 2. Now take the limit as h → 0.

Note: If you have simplified the expression correctly, you can often take the limit by just setting h = 0. (See the discussion in Section 3.6 of Applied Calculus for more details.)

Example Continued The derivative, f'(x), is given by:

f'(x) =

f'(1) =

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

The rupiah is the Indonesian currency

where t is time in days. (t = 0 represents noon on Monday.)

R(t) =
R(t+h) =
R(t+h) - R(t) =
 \frac{R(t+h) - R(t)}{h} =

 R'(t) = limh→0 \frac{R(t+h) - R(t)}{h} =

This quantity is measured in
 At noon on , the dollar was rising at a rate of rupiah per day.

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} =

Q Finally, f'(x) =

Now try some of the exercises in Section 3.6 in Applied Calculus or Section 10.6 in Finite Mathematics and Applied Calculus.

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Last Updated: June, 2008