Find
𝑑
𝑑
𝑥
[
cot
⁡
(
𝑥
)
]
.

Question

Find 
𝑑
𝑑
𝑥
[
cot
⁡
(
𝑥
)
]
.

BlackTom AI · Accepted Answer

The problem asks for the derivative of cot(x) with respect to x across the given choices.

Option 1: d/dx [cot(x)] = csc(x) cot(x). This is incorrect because the derivative of cot(x) is not their product; that expression resembles the derivative of csc(x) (which is -csc(x) cot(x)) moved around, but it does not match the cot derivative rule.

Option 2: d/dx [cot(x)] = - csc(x) cot(x). This is also incorrect for cot(x); -csc(x) cot(x) is the derivative of csc(x), not cot(x). Confusing the roles of the two basic trig functions leads to this misstep.

Option 3: d/dx [cot(x)] = - csc^2(x). This is the standard and correct result. It can be derived directly by writing cot(x) = cos(x)/sin(x) and applying the quotient rule: 
- The numerator derivative: (-sin(x)) * sin(x) - cos(x) * cos(x) = -(sin^2(x) + cos^2(x)) = -1. 
- The denominator is sin^2(x), so the derivative is -1 / sin^2(x) = -csc^2(x).

Option 4: d/dx [cot(x)] = + csc^2(x). This is incorrect due to the sign; the negative sign arises from the derivative of sine in the quotient rule, so the result must be negative, not positive.

In summary, the only expression that matches the correct derivative d/dx [cot(x)] is -csc^2(x); the other options mix up derivative identities or signs, leading to incorrect results.

MATH_1225_17255_202501 3.3 Derivatives of Trigonometric Functions

Find 𝑑 𝑑 𝑥 [ cot ⁡ ( 𝑥 ) ] .

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