Main Page Everything for Finite Math Everything for Applied Calc Everything Topic Summaries On Line Tutorials On Line Utilities               ← Previous Summary Next Summary → Tutorial for This Topic Review Exercises E-mail Webmaster Textbook Lleveme a la Pagina Español Applied calculus topic summary: introduction to the derivative Tools: Function Evaluator & Grapher | Excel Grapher | Excel First and Second Derivative Grapher Subtopics: Average Rate of Change | Derivative | Numerical Approach | Geometric Approach | Algebraic Approach | Velocity

Average Rate of Change of f(x) over [a, b]: Difference Quotient

The average rate of change of f(x) over the interval [a, b] is

 Average rate of change = Δf Δx = f(b) - f(a) b - a .
We also call this average rate of change the difference quotient of f(x) over the interval [a, b]. Its units of measurement are units of f(x) per unit of x.
Alternative Formulation: Average Rate of Change of f(x) over [a, a+h]
(Replace b above by a+h.)

The average rate of change of f(x) over the interval [a, a+h] is

 Average rate of change = f(a+h) - f(a) h .

Units: The units of the average rate of change are units of f per unit of x.

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Example

Let f(x) = 2x2 - 4x + 1. Then the average rate of change of f(x) over the interval [2, 4] is

Average rate of change = f(4) - f(2) 4 - 2
= 17 - 1 2 = 8

Interpretation: If, say f(x) represents the annual profit of your company (in millions of dollars) and x represents the year since January 2003, then the units of measurement of the average rate of change are millions of dollars per year. Thus, your company made an average annual profit of \$8 million per year over the period January 2005 (t = 2) to January 2007 (t = 4).

Use the handy little utility below to compute the average change of the above function f(x) over other intervals. Enter the x-coordinates (a and b in the formula), leave everything else blank, and press "Compute." (You can also change the function to anything you like, using standard technology formatting.)

 f(x) = a =     b = Ave. Rate of Change:

You can also use the function evaluator to compute average rates of change.

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Instantaneous Rate of Change of f(x) at x = a: Derivative

The instantaneous rate of change of f(x) at x = a is given by taking the limit of the average rates of change (computed by the difference quotient) as h approaches 0.

 Instantaneous rate of change = limh→0 f(a+h) - f(a) h .
We also call this instantaneous rate of change the derivative of f(x) evaluated at x = a, and write it as f'(a) (read "f prime of a"). Its units of measurement are units of f(x) per unit of x. Thus,
 f'(a) = limh→0 f(a+h) - f(a) h .

Note:

• f'(a) = Instantaneous rate of change of f at the point a.
• f'(x) = Instantaneous rate of change of f at the point x.
Hence, the derivative f'(x) is a function of x.

Since f'(x) is a limit, it may or may not exist. That is, the quantities [f(x+h) - f(x)]/h may or may not approach a fixed number as h approaches zero. If everything works out fine and the limit exists, then we say that f is differentiable at x. Otherwise, we say that f is not differentiable at x.

On this page, we summarize three ways of obtaining the derivative of a function at a point: numerical, graphical, and algebraic.

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Examples

Let f(x) = 2x2 - 4x + 1, as above. Then the instantaneous change of f(x) at x = 2 is

f'(2) = 4.
(We shall see where this answer came from below.)

Interpretation
If, say f(x) represents the annual profit of your company (in millions of dollars) and x represents the year since January 2003, then the units of measurement of the instantaneous rate of change are millions of dollars per year. Thus, your company's annual profit was increasing at a rate of \$4 million per year at the start of 2005 (t = 2).

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Numerical Approach

To compute an approximate value of f'(a) (for a given value of a) numerically, one can use either:

1. A table of values
2. A quick approximation
The first approach shows better and better approximations, sometimes allowing you to guess the exact value, while the second method gives a quick estimate.

Using a Table
In a table, compute a succession of values of difference quotients

 f(a+h) - f(a) h
for smaller and smaller values of h, and decide what number these values are approaching. (See the example opposite.)

A Quick Approximation
Use a single small value of h and compute the difference quotient:

 f'(a) ≈ f(a + 0.0001) - f(a) 0.0001 Forward Difference Quotient
Here, we chose h = 0.0001. The smaller h, the better the approximation. (See the example opposite.)

Another Quick Approximation: Balanced Difference Quotient
The following formula often gives a better estimate of the derivative

 f'(a) ≈ f(a+0.0001) - f(a-0.0001) 0.0002 Balanced Difference Quotient

Derivative Calculator (Balanced Difference Quotient Approximation)
Enter a function, enter the point a, adjust h as you want, and press "Compute".

 f(x) = a =     h = f'(a) Top of Page
Examples

Continuing with the example f(x) = 2x2 - 4x + 1, let us compute an approximate value of f'(2).

Using a Table: The difference quotient (with a = 2) is

 f(2+h) - f(2) h
=
2(2+h)2-4(2+h)+1 - (2(2)2-4(2)+1) h
The following table shows the value of this difference quotient for several values of h.

 h 1 0.1 0.01 0.001 Difference Quotient 6 4.2 4.02 4.002

As h gets smaller, we see that the value gets closer and closer to 4, so we conclude

f'(2) ≈ 4.

Using A Quick Approximation (Forward Difference Quotient): We use the formula (with a = 2)

 f'(2) ≈ f(2+0.0001) - f(2) 0.0001 = f(2.0001) - f(2) 0.0001 = 1.00040002 - 1 0.0001 = 4.0002.
(You could have used the little utility at the top of the page to do this calculation.)

Notice that the "quick approximation" method does not give the exact answer, but the balanced difference quotient will in this case (see opposite).

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Geometric Approach: The Derivative as Slope

Secant and Tangent Lines
The slope of the secant line through the points on the graph of f where x = a and x = a+h is given by the slope of the line PQ in the following diagram: msec = Slope of secant line through P and Q = f(a+h) - f(a) h

This is also the formula for the average rate of change of f over [a,a+h]. So,

Slope of Secant = Average rate of change

The slope of the tangent line through the point on the graph of f where x = a is obtained by moving the point Q closer to P; in other words, by letting h approach 0: mtan = slope of tangent = limh→0 f(a+h) - f(a) h = f'(a)

This is also the formula for the instantaneous rate of change of f at the point a. So,

Slope of Tangent = Instantaneous rate of change = Derivative

We can approximate the slope of the tangent through the point where x = a by using the balanced difference quotient,

 mtan ≈ f(a+0.0001) - f(a 0.0001) 0.0002 .

Zooming In
We can also interpret the derivative, or slope of the tangent, at a given point on the graph as the slope of the (almost) straight line obtained by "zooming-in" to that point on the curve.(See opposite.)

Extra Topic: Graph of the Derivative
Press here for on-line on how to obtain the graph of f' from the graph of f.

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Examples

Continuing with the example f(x) = 2x2 - 4x + 1,

 Slope of secant line through points where x = 2 and x = 3 = Average rate of change of f(x) over [2, 3] = 6   (see calculation in the table above)
 Slope of tangent line through point where x = 2 = Instantaneous rate of change of f(x) at x = 2 = 4   (see quick approximation above)

Here is the graph with these two lines shown: Zooming In

Here is an illustration of zooming in to a point on a graph where x = 0.75. Top of Page
Computing the Derivative Algebraically

To compute the derivative of a function algebraically, proceed as follows.

1. Write down the definition of the derivative,
 f'(x) = limh→0 f(x+h) - f(x) h .
2. Substitute for f(x+h) and f(x)
You can use an actual value for x if you are asked, say, to compute f'(3), or just leave it as x if you are asked for the derivative function f'(x) .
3. Simplify the numerator in order to factor out an "h." Then cancel the "h"s and take the limit to obtain the answer.
Sometimes, you need to do more than just simplify the numerator...

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Example

Going back to our first example, f(x) = 2x2 - 4x + 1, let us now calculate f'(x) algebraically by following the steps in the adjacent window.

f'(x) =  limh→0 f(x+h) - f(x) h
=  limh→0 (2(x+h)2-4(x+h)+1) - (2x2-4x+1) h
=  limh→0 2x2+4xh+2h2-4x-4h+1-2x2+4x-1 h
=  limh→0 4xh+2h2-4h h
=  limh→0 h(4x+2h-4) h
=  limh→0 (4x+2h-4)
=4x-4

Thus,  f'(x) = 4x-4.

Go to the tutorial on average rates of change for practice in computing average rates of change algebraically (what we did above up to the last step), or to tutorial on computing the derivative algebraically and scroll down to the box called "Computing the Derivative Algebraically." The "Help" button brings up the complete solution.

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Velocity

For an object moving in a straight line with position s(t) at time t, the average velocity from time t to time t+h is given by the difference quotient

 vaverage = s(t+h) - s(t) h .

The instantaneous velocity at time t is given by

 v(t) = s'(t) = limh→0 s(t+h) - s(t) h .

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Examples

Suppose the position of a moving object is given by

s(t) = t2 -2t+4 miles
at time t hours. Then its velocity at time t is given by
s'(t) = 2t-2 miles per hour.
Thus, for example, its velocity at time t = 3 hours is
s'(3) = 4 miles per hour.

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Last Updated: April 2010