Find the Derivative - d/dx csc(x)

Problem

d(csc(x))/d(x)

Solution

1. Rewrite the cosecant function in terms of the sine function using the reciprocal identity csc(x)=1/sin(x)

2. Apply the power rule and the chain rule by treating the expression as (sin(x))(−1)

3. Differentiate the expression to get −1*(sin(x))(−2)⋅d(sin(x))/d(x)

4. Substitute the derivative of sin(x) which is cos(x) into the expression to get −cos(x)/sin2(x)

5. Separate the fraction into two parts: −1/sin(x)⋅cos(x)/sin(x)

6. Simplify using the trigonometric identities csc(x)=1/sin(x) and cot(x)=cos(x)/sin(x)

Final Answer

d(csc(x))/d(x)=−csc(x)*cot(x)

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