Derivatives -- Determining the Derivative of a Function as a Limit

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

The derivative f'(x) of a function f(x) is defined as either of the limits

                f(s) - f(x)
f'(x) =   lim   -----------
        s --> x    s - x


                f(x + h) - f(x)
f'(x) =   lim   ---------------
        h --> 0         h

This definition is important for two reasons.

It tells us how to estimate the derivative.
It enables us to compute the exact derivative in many cases.

In this module we look at the second use.

Example:

Find the derivative of the function f(x) = x^2.

Answer 1:

Missing graphic

We give two answers.

Answer 2:

Missing graphic

Use your CAS window to graph the functions f(x) = x^2 and, as determined above, f'(x) = 2x. Does f'(x) look like you expect the derivative of f(x) = x^2 to look?

Because the derivative of a function is the limit of quotients whose numerator comes from values of the function and whose denominator comes from values of the independent variable, the units in which the derivative is measured are the units in which values of the function are measured divided by the units in which the independent variable is measured. For example, if the function f(t) represents the number of gallons of water in a container at time t and the time t is measured in minutes then f'(t) is measured in gallons per minute. See the module on units for more.

Check Your Understanding

For each of the following problems do the following.

Find the derivative of the function f(x) by finding one of the two limits.
Graph the function f(x) and the derivative f'(x) as determined above and check that the derivative appears as expected.

f(x) = 1/x.
answer
f(x) = x^3 - x.
answer
f(x) = m x + b where m and b are two constants.
answer
f(x) = ax^2 + bx + c where a, b, and c are constants.
answer
f(x) = sin x. Hint: See the module Estimation and Limits -- Some Limits Involving Sine and Cosine.
answer
f(x) = cos x.
answer

Challenge Problem

In most mathematics classes functions are usually described by formulas -- for example,

f(x) = sin 2 x

In the real world, however, functions are often given by some sort of physical process. For example, a chemist might perform an experiment in which she mixes a solution with a certain concentration, x, of a chemical and then makes some measurement f(x) that depends on the concentration x. The concentration x is the input to this experiment or function. The measurement f(x) is the output or value of the function with input x. Similarly, a market researcher for a chain of stores might run an experiment in which he priced a particular product at p dollars and then measured the number f(p) of units sold at the price p. The price p is the input to this experiment or function. The number, f(p), of units sold is the output or value of the function with input p.

When a function is determined in this way it cannot be computed as precisely as a function defined by some mathematical formula. The instruments used to measure physical quantities like temperature and mass only measure these quantities to within a certain precision. The more precision one needs, the more expensive the instrument. The JAVA applet below simulates a function determined by some physical experiment. You can choose any value that you want for the independent variable or input by typing it in the upper box. Then run the experiment by clicking the button. The result will appear below the button.

The output will have six digits that are correct. For example, the output 1.23456 would mean that the actual value is 1.23456 after rounding to six digits. The output 1.2345 is the same as 1.23450.

Challenge Problem:

Find the best estimate that you can for the derivative of the function given by the JAVA applet above for the input 1.