𝑑
𝑑
𝑥
[
(
sin
⁡
(
𝑥
)
)
−
1
]
=

Question

BlackTom AI · Accepted Answer

We start by identifying the function inside the derivative: d/dx [ (sin(x))^{-1} ], which is the derivative of csc(x) since csc(x) = 1/sin(x).
Option 1: − cot(x) csc(x). This is the standard derivative result: d/dx [ csc(x) ] = − csc(x) cot(x). Since csc(x) = 1/sin(x) and cot(x) = cos(x)/sin(x), multiplying gives − (1/sin(x)) * (cos(x)/sin(x)) = − cos(x) / sin^2(x), which matches − csc(x) cot(x). Therefore this option correctly represents the derivative.
Option 2: arcsin(x). This would be the derivative of arcsin(x), not of csc(x) or its reciprocal. It has no relation to the derivative of 1/sin(x) and is therefore incorrect here.
Option 3: 1 / (1 − x^2). This expression resembles the derivative of atanh(x) or a variant of a partial fraction form, but it is unrelated to the derivative of csc(x). It does not match d/dx [ (sin x)^{-1} ]. Hence it is incorrect.
Option 4: − 1 / cos^2(x). This equals − sec^2(x). While it is a derivative form encountered in d/dx [ tan(x) ] or d/dx [ sec(x) ], it does not equal the derivative of csc(x). Therefore this option is incorrect.

MATH_1225_17255_202501 3.5 Implicit Differentiation

𝑑 𝑑 𝑥 [ ( sin ⁡ ( 𝑥 ) ) − 1 ] =

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