Page 98 - Calculus Workbook For Dummies

82 Part III: Differentiation

Solutions for Differentiation Problems

a f x = 8 ; f l ^ h 0
x =
^ h
The derivative of any constant is zero.
b g x = π 3 ; g x = 0
^ h
l ^ h
Don’t forget that even though π sort of looks like a variable (and even though other Greek let-
ters like ,i a , and ~ are variables), π is a number (roughly 3.14) and behaves like any other
number. The same is true of e . . 2 718. And when doing derivatives, constants like c and k also
behave like ordinary numbers.

c g x = k sin π cos π (where k is a constant); g x = 0
2
l ^ h
^ h
2
If you feel bored because this first page of problems was so easy, just enjoy it; it won’t last.
d f x = 5 x 4 ; f l ^ h 20 x 3
x =
^ h
Bring the 4 in front and multiply it by the 5, and at the same time reduce the power by 1, from 4
3
x =
to 3: f l ^ h 20 x . Notice that the coefficient 5 has no effect on the derivative in the following
4
3
sense: You can ignore the 5 temporarily, do the derivative of x (which is 4x ), and then put the
5 back where it was and multiply it by 4.
e g x = - x 3 ; g x = - 3 x 2
l ^ h
^ h
10 10
- 3 x 2
You can just write the derivative without any work: 10 . But if you want to do it more method-
ically, it works like this:
- x 3 1
3
1. Rewrite so you can see an ordinary coefficient: - x .
10 10
2. Bring the 3 in front, multiply, and reduce the power by 1.
- 3 2 - 3 x 2
g x = x (which is the same, of course, as .)
l ^ h
10 10
5
f y = x - 5 ^ x $ 0h ; y =- x - / 7 2
l
2
Rewrite with an exponent and finish like problem 5 ` x - 5 = x - / 5 2 . j
5 - 5
l
To write your answer without a negative power, you write y = - or / 7 2 . Or you can write
2 x / 7 2 2 x
5 - 5 5 - 5
l
your answer without a fraction power, to wit: y = - or or - 7 or 7 .
2 x 7 2 x 7 2` xj 2` xj
You say “po-tay-to,” I say “po-tah-to.”
g s t = t 7 + + 10 ; s t = 42 t + 1
5
6
t
l ^ h
^ h
Note that the derivative of plain old t or plain old x (or any other variable) is simply 1. In a
sense, this is the simplest of all derivative rules, not counting the derivative of a constant. Yet
for some reason, many people get it wrong. This is simply an example of the power rule: x is the
1
same as x , so you bring the 1 in front and reduce the power by 1, from 1 to 0. That gives you
0
1x . But because anything to the 0 power equals 1, you’ve got 1 times 1, which of course is 1.
2
3
5
h y = _ x - 6i ; y = l 6 x - 36 x 2
FOIL and then take the derivative.
3
3
= _ x - 6 _i x - 6i
3
6
= x - 12 x + 36
5
y = l 6 x - 36 x 2