Find dy/dx y=(cot(x))/(1+cot(x))

Problem

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

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two expressions, apply the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

2. Assign the variables for the numerator and denominator. Let u=cot(x) and v=1+cot(x)

3. Differentiate both components with respect to x

d(cot(x))/d(x)=−csc^2(x)

d(1+cot(x))/d(x)=−csc^2(x)

4. Substitute these derivatives into the quotient rule formula.

d(y)/d(x)=((1+cot(x))*(−csc^2(x))−cot(x)*(−csc^2(x)))/((1+cot(x))^2)

5. Distribute the terms in the numerator.

d(y)/d(x)=(−csc^2(x)−cot(x)*csc^2(x)+cot(x)*csc^2(x))/((1+cot(x))^2)

6. Simplify the numerator by canceling the opposite terms −cot(x)*csc^2(x) and cot(x)*csc^2(x)

d(y)/d(x)=(−csc^2(x))/((1+cot(x))^2)

Final Answer

d(y)/d(x)=(−csc^2(x))/((1+cot(x))^2)

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