Page 184 - Calculus Demystified

CHAPTER 6
SOLUTION Transcendental Functions 171
We take the natural logarithm of both sides:
(sin x)
ln F(x) = ln (cos x) = (sin x) · (ln(cos x)). (†)
Now we calculate the derivative using the formula (∗) preceding this example:
The derivative of the left side of (†) is

F (x)
.
F(x)
Using the product rule, we see that the derivative of the far right side of (†) is

− sin x
(cos x) · (ln(cos x)) + (sin x) · .
cos x
We conclude that

F (x) − sin x
= (cos x) · (ln(cos x)) + (sin x) · .
F(x) cos x
Thus
! "
2
sin x

F (x) = (cos x) · (ln(cos x)) − · F(x)
cos x
! 2 "
sin x
= (cos x) · ln(cos x) − · (cos x) (sin x)
cos x
You Try It: Differentiate log | cos x|.
9
You Try It: Differentiate 3 sin(3x) . Differentiate x sin 3x .
EXAMPLE 6.29
2
x
Calculate the derivative of F(x) = x · (sin x) · 5 .
SOLUTION
We have
2 x
[ln F(x)] =[ln(x · (sin x) · 5 )]

=[(2 · ln x) + ln(sin x) + (x · ln 5)]
2 cos x
= + + ln 5.
x sin x
Using formula (∗), we conclude that

F (x) 2 cos x
= + + ln 5
F(x) x sin x