Mathematics

Mathematics

Problem

Which statement best accounts for the negative sign in the derivative formula ddx(csc⁡x)=−csc⁡xcot⁡x\dfrac{d}{dx}\bigl(\csc x\bigr)=-\csc x\cot x ? a) The derivative of sin⁡x\sin x is negative, so the minus sign appears in the derivative of its reciprocal. b) Applying the quotient rule to 1sin⁡x\displaystyle\frac{1}{\sin x} gives sin⁡x−cos⁡xsin⁡2x\dfrac{\sin x-\cos x}{\sin^2 x}; the subtraction produces the negative sign. c) Differentiating (sin⁡x)−1(\sin x)^{-1} multiplies the result by −1-1 because the exponent is −1-1. d) Since csc⁡x\csc x is the inverse trigonometric function of sin⁡x\sin x, the derivative must contain a negative factor.

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