D. P. Morstad, University of North Dakota
Objectives of Assignment
1.
To familiarize you with DIFF’s interface.
2. To provide practice identifying the rules of differentiation in examples which systematically apply the rules of differentiation.
I. Introduction
The software program Differential Calculus, v3.1, (DIFF) performs symbolic differentiation in a very systematic and step–by–step manner. Each step is very logical, and
DIFF only takes very small steps. If differentiation is still a little bit confusing to you, seeing
DIFF’s steps can help simplify and demystify it.
DIFF is very easy to use and get answers from. However, the value of DIFF is the process it uses, not the final answers it finds. Looking at each step will help you examine the process of differentiation in microscopic detail.
II. How to Use DIFF
Double-click on the Diff icon. When DIFF has loaded, you will notice that it looks somewhat like X(PLORE), except that it is more colorful and a little busier on the screen. In the upper right of the screen is the list of keystrokes that make DIFF perform specific tasks.
The “^” mark means to hold down the
F key while simultaneously depressing the other indicated key.
Example 1. How DIFF finds the derivative of a sum: finding
+
3 )
. dx
Solution: Two simple steps.
1) Type “ x + 3” to enter the function. It should appear in the middle of the screen.
2) Hold down the
F
key and press d
to make DIFF find the derivative.
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The screen should have “ x + 3” at the top and “1” at the bottom. These are the original function and the derivative. Simple enough. The important stuff is what’s in between.
Let’s analyze this stuff. Each step taken by DIFF is listed below. The italics following each step is a brief explanation of the rationale for the step.
= x + 3 This is the original function you typed in.
D[ x
= D[
+ 3] x ] + D[3]
This means DIFF wants to find the derivative of (x + 3).
DIFF knows that to find the derivative of two things added together, you just find the derivative of each part separately.
This is the addition rule.
The derivative of x is 1, and the derivative of 3 is 0.
= [1] + [0]
= 1 1 + 0 is 1. This is the final answer.
Certainly this is an absurd amount of work just to find the derivative of such a simple function, but the point is to demonstrate the rigorously systematic nature of DIFF. For your homework you will have to provide the rationale for each of DIFF’s steps in a variety of problems. However, you will not have to write in italics.
Example 2. How DIFF finds the derivative of a product. Find d ( 4 x )
. dx
Solution: First press any key, and then press
X
to get back to the work screen.
Use the
Q
(backspace key) to remove the previous example.
1) Type “4 [ x ”. (The “ [ ” is necessary to indicate multiplication.)
2) Hold down the
F
key and press d
to make DIFF find the derivative.
Each step used by DIFF is listed below, followed by the rationale.
= 4 [ x The original function.
What needs to be differentiated . D[4 [ x ]
= [4] [ D[ x ] + [x] [ D[4] This is just the product rule. At first, DIFF does not distinguish between functions and numbers. All it sees is one quantity times another, so it knows it must use the product rule.
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= [4]
= 4
[ [1] + [ x ] [ [0] DIFF found the derivative of x and the derivative of 4.
Simplification yields the correct answer .
Again, notice how rigidly systematic DIFF is. You and I both know that using the product rule here just turned an easy problem into a pain, but DIFF follows the rules religiously.
You need to know this in order to decipher DIFF’s individual steps in more complicated problems.
Example 3. A function with a product and a sum. Find: d ( 5 x
−
1 )
. dx
Solution: The output and rationale follows.
= 5 [ x – 1
D[5 [ x ]
Original function.
The first small individual part that needs to be differentiated by itself (the first term).
= [5]
[
D[ x ] + [ x ]
[
D[5] Applied the product rule to D[5
[ x].
= [5]
[
[1] + [ x ]
[
[0] Found derivatives of x and 5.
= 5
D[5
[ x – 1]
Simplified.
Now differentiate while including the next term.
= D[5
[ x ] – D[1]
= [5] – [0]
Applied the addition rule.
DIFF already figured out D[5
[ x] above, so it just used substitution and then also found the derivative of
1.
The final answer. = 5
DIFF breaks differentiation into very small parts and then applies the simple rules of differentiation. It is this ability to analyze the function and then break it into smaller parts which you should try to assimilate and learn. d ( 3 x
2
)
Example 4. The power rule and the chain rule. Find:
Solution:
.
Type “3 [ x ^2” and press dx
Fd
. Steps and rationale follow.
= 3 [ x ^2 Original function.
D[ x ^2] DIFF goes as far “inside” the function as it can in order to find what to differentiate first.
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= 2
= 2
[
[
{[
{[ x x
]^[2 – 1]}
]^[2 – 1]}
[
[
D[
1 x ] DIFF uses the exponent rule to bring down the exponent and to subtract 1 from the exponent.
DIFF is unsure whether or not x is an inside function, so it applied the chain rule. That’s why
“D[x]” is multiplied at the end.
Differentiated x.
= 2 [ x Algebraically simplified. (This is just the derivative of the x^2 part.)
Includes the next part of the function. D[3 [ x ^2]
= [3] [ D[ x ^2] + [ x ^2] [ D[3] Applied the product rule.
= [3]
= 3 [
[
2
[2
[ x
[ x ] + [ x ^2] [ [0] Substituted the derivative of x^2 which was already found above, and found the derivative of 3.
Algebraically simplified.
= 6 [ x Final answer.
Example 5. The quotient rule. Find: d

 x x
+
−
1 
1

. dx
Solution: Type “( x + 1)/( x – 1)” and press
Fd .
= ( x + 1)/( x – 1) Original function.
D[ x + 1]
= D[ x ] + D[1]
= [1] + [0]
= 1
D[x – 1]
= D[x] – D[1]
= [1] – [0]
First find the derivative of the numerator .
Applied the addition rule.
Found simple derivatives .
This is the derivative of the numerator .
Also need to find the derivative of the denominator .
Applied the addition rule .
Evaluated simple derivatives .
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= 1
D[( x + 1)/( x – 1)]
Derivative of the denominator .
Now find the derivative of the entire quotient .
= {[ x – 1] [ D[ x + 1] – [ x + 1] [ D[ x – 1]}/[ x – 1]^2
= {[ x – 1] [ [1] – [ x + 1] [ [1]}/[ x – 1]^2 numerator and denominator.
Applied quotient rule.
Substituting previously found derivatives of
= ( x – 1 – ( x + 1) ) / ( x – 1)^2 Algebraically simplified.
= –2/( x – 1)^2 Algebraically simplified to the final answer.
III. Practice Exercises
In these 12 exercises, have DIFF find the derivative of each of the following functions.
For each one, write down the rationale for every step that DIFF takes. Do this just like the example problems above.
1. f x
=
( 3 x
−
1 )
2
2. f x
=
( x
2 +
3. f x
=
( 3 x
−
5)
−
3
4. f x
=
( 3 x
+
2 )
4
5. f x
=
( x
2 +
1 )
1
2 6. f x
= x x
+
1
7. f x
= x x
2 − x ) 8. f x
= x x
+
2 )
5
9. f x
= x x
3 +
5) 10. f x
=
(( x
+
2
11. f x
= x x
2
+
3
12. f x x x
3
6
13. f ( x )
== cos
2
( 3 x
++
1 ) 14. f ( x )
== sin
2 cos x
++ cos
2 cos x
(hint: type cos(x)^2) (hint: see the hint for #13)
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IV. Further Capabilities of DIFF (optional)
Graphing
It is often useful and informative to examine the graph of a function and its derivative plotted on the same axis system. After DIFF finds a derivative, it will plot the original function and the derivative.
Enter the function f x
= x 2 and have DIFF find the derivative. Then, to have DIFF graph both, press
Fg
. This brings up the graphing information screen. Use the down cursor arrow key to move down to the x minimum field - it’s probably 0. Type –6 and press
E to set the minimum value of x at –6. For x maximum, type 6 and press
E
. Use the cursor arrow to move to the y minimum and maximum values. Use –6 and 6 again. Now press
Fg
again. You should see a parabola (the original function, y = x
2
) and a dotted straight line
(the derivative, y = 2 x ).
Go back to the input screen and have DIFF differentiate y
= x press
Fg
twice. You do not have to reset the graphing information. You should see the sine function (the original) and the cosine function (its derivative).
Evaluating
Once DIFF has found the derivative, it can numerically evaluate the function and the derivative. It displays the results in a table.
For example, suppose you want a table of values for the function f x
= x
2
and its derivative from x = –4 to x = 4. To do this, enter the function and have it differentiated. Get back to the input screen. Then press
Fy
. Type –4, press
E
; then 4, press
E
; and then 8, press
E
.
Higher Derivatives
Sometimes it is necessary to differentiate a function twice, three times, or even more times. DIFF can quickly do this by moving the derivative up into the original function window.
For example, suppose you want to find the third derivative of f x
= x . Enter the function and DIFF differentiate it. Get back to the input screen. You can see that the first
1
2 derivative is , or, in simplified form, sec x . cos 2 x
Now press
Fw
. This moves the derivative up into the function window. Simply press
Fd
to find the second derivative. It should be
2 sin cos 3 x x
, or, in simplified form,
2 sec
2 x tan x . press
Press
Fw
once more to get the second derivative into the function window. Then
Fd
finally get the third derivative,
2 (cos 4 x
+
3 x x cos 6 x
. Go ahead, simplify it.
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