Find the Derivative - d/dx y=(1+csc(x))/(1-csc(x))

Problem

d()/d(x)(1+csc(x))/(1−csc(x))

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two expressions, use the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions. Let u=1+csc(x) and v=1−csc(x)

3. Differentiate the individual components. The derivative of csc(x) is −csc(x)*cot(x)

d(u)/d(x)=−csc(x)*cot(x)

d(v)/d(x)=csc(x)*cot(x)

4. Substitute these into the quotient rule formula.

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

5. Factor out the common term −csc(x)*cot(x) from the numerator to simplify the expression.

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

6. Simplify the terms inside the brackets. The −csc(x) and +csc(x) cancel out, leaving 1 + 1 = 2$.

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

Final Answer

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

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