Which of these is the result when you differentiate the expression
ln
⁡
(
3
cos
⁡
(
𝑥
)
)
with respect to
𝑥
?

Question

Which of these is the result when you differentiate the expression 
ln
⁡
(
3
cos
⁡
(
𝑥
)
)
 with respect to 
𝑥
?

BlackTom AI · Accepted Answer

1. Relevant knowledge points and concepts
The derivative of ln(u(x)) is u'(x)/u(x) (chain rule). The derivative of cos x is −sin x. You may also use ln(ab) = ln a + ln b to separate constants. Domain note: ln(3 cos x) is defined only when 3 cos x > 0, i.e., cos x > 0, but the differentiation rule applies on intervals where the function is defined.

2. Step-by-step reasoning
Let u(x) = 3 cos x. Then d/dx[ln(3 cos x)] = u'(x)/u(x) by the derivative rule for natural log.

3. Calculations with intermediate steps
u(x) = 3 cos x
u'(x) = 3 · (d/dx cos x) = 3 · (−sin x) = −3 sin x
So d/dx[ln(3 cos x)] = u'(x)/u(x) = (−3 sin x)/(3 cos x)

Simplify by canceling the factor 3:
(−3 sin x)/(3 cos x) = − sin x / cos x = − tan x

Alternative route: ln(3 cos x) = ln 3 + ln(cos x). d/dx[ln 3] = 0 and d/dx[ln(cos x)] = (−sin x)/cos x = −tan x, giving the same result.

4. Why the correct answer is correct
Applying the standard derivative of ln(u) together with the derivative of cos x yields (−3 sin x)/(3 cos x). The numeric factor 3 cancels, leaving −sin x / cos x, which is exactly −tan x. This is the simplified and correct derivative.

5. Why other options are incorrect
No other answer choices are listed here, but common incorrect responses include tan x (wrong sign), sec^2 x (derivative of tan x, not ln of cos), or −sin x (missing division by cos x). These do not follow the ln chain rule and simplification above.

6. Conclusion
The derivative of ln(3 cos x) is −tan x, which matches the provided correct answer.

Which of these is the result when you differentiate the expression ln ⁡ ( 3 cos ⁡ ( 𝑥 ) ) with respect to 𝑥 ?单项选择题

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