Find the Derivative - d/dx f(x)=csc(x)

Problem

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

Solution

1. Identify the function to be differentiated as the cosecant function, which can be expressed in terms of sine as csc(x)=1/sin(x)

2. Apply the reciprocal rule or the power rule by rewriting the expression as (sin(x))(−1)

3. Use the chain rule to differentiate the expression.

4. Differentiate the outer function to get −1*(sin(x))(−2) and multiply by the derivative of the inner function, which is cos(x)

5. Simplify the resulting expression by grouping the trigonometric terms.

−cos(x)/sin2(x)

6. Rewrite the expression using standard trigonometric identities where 1/sin(x)=csc(x) and cos(x)/sin(x)=cot(x)

Final Answer

d(csc(x))/d(x)=−csc(x)*cot(x)

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