Derivative: Algebraic Viewpoint

3.6 Derivatives: 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) 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 rate of change of f at x. Thus,

f'(x)

Slope of Tangent at x

Rate of Change of f at x

lim
h

f(x+h) - f(x)

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,

f(x+h) - f(x)

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) = 3x² + 4x. Enter the required expressions using valid technology format.

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 3.3 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 the value of the Dollar at 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:

Q Let f(x) = 1

x .

f(x+h) is given by

x + h

undefined

Q f(x+h) - f(x)

h =

-1

x(x+h)

-h²

x(x+h)

h²

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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