If y = csc(3x), dy/dx equals?

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Multiple Choice

If y = csc(3x), dy/dx equals?

Explanation:
The derivative of cosecant uses the chain rule: d/dx [csc(u)] = -csc(u) cot(u) · u'. For y = csc(3x), let u = 3x, so u' = 3. Multiply: dy/dx = -csc(3x) cot(3x) · 3 = -3 csc(3x) cot(3x). This negative sign comes from the derivative of cosecant, and the factor 3 comes from the inner derivative of 3x. So the correct expression is -3 csc(3x) cot(3x). The other forms would miss the negative sign, replace csc with sec, or omit the inner derivative of 3x.

The derivative of cosecant uses the chain rule: d/dx [csc(u)] = -csc(u) cot(u) · u'. For y = csc(3x), let u = 3x, so u' = 3. Multiply: dy/dx = -csc(3x) cot(3x) · 3 = -3 csc(3x) cot(3x). This negative sign comes from the derivative of cosecant, and the factor 3 comes from the inner derivative of 3x. So the correct expression is -3 csc(3x) cot(3x). The other forms would miss the negative sign, replace csc with sec, or omit the inner derivative of 3x.

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