Page 86 - Calculus Demystified

Foundations of Calculus
CHAPTER 2
Practice is the essential tool in mastery of these ideas. Be sure to do all the You Try 73
It problems in this section.
EXAMPLE 2.14
Calculate the derivative

d 3
[(sin x + x) · (x − ln x)].
dx
SOLUTION
2
3
We know that (d/dx) sin x = cos x, (d/dx)x = 1, (d/dx)x = 3x , and
(d/dx) ln x = (1/x). Therefore, by the addition rule,
d d d
(sin x + x) = sin x + x = cos x + 1
dx dx dx
and
d 3 d 3 d 2 1
(x − ln x) = x − ln x = 3x − .
dx dx dx x
Now we may conclude the calculation by applying the product rule:

d
3
(sin x + x) · (x − ln x)
dx
d d
3
3
= (sin x + x) · (x − ln x) + (sin x + x) · (x − ln x)
dx dx

1
3 2
= (cos x + 1) · (x − ln x) + (sin x + x) · 3x −
x

1
3 3 2
= (4x − 1) + x cos x + 3x sin x − sin x − (ln x cos x + ln x).
x
EXAMPLE 2.15
Calculate the derivative
x
d e + x sin x
.
dx tan x
SOLUTION
x
x
We know that (d/dx)e = e , (d/dx)x = 1, (d/dx) sin x = cos x, and
2
(d/dx) tan x = sec x. By the product rule,

d d d
[x · sin x]= x · sin x + x · sin x = 1 · sin x + x · cos x.
dx dx dx