Find the Derivative - d/dx (cot(x))/(1+cot(x))

Problem

d()/d(x)cot(x)/(1+cot(x))

Solution

1. Identify the rule needed for differentiation, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions: u=cot(x) and v=1+cot(x)

3. Differentiate both parts: d(cot(x))/d(x)=−csc2(x) and d(1+cot(x))/d(x)=−csc2(x)

4. Substitute these into the quotient rule formula:

((1+cot(x))*(−csc2(x))−cot(x)*(−csc2(x)))/((1+cot(x))2)

5. Distribute the terms in the numerator:

(−csc2(x)−cot(x)*csc2(x)+cot(x)*csc2(x))/((1+cot(x))2)

6. Simplify the numerator by canceling the terms cot(x)*csc2(x) and −cot(x)*csc2(x)

(−csc2(x))/((1+cot(x))2)

Final Answer

d()/d(x)cot(x)/(1+cot(x))=−csc2(x)/((1+cot(x))2)

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