Find the Derivative - d/dx (cot(x))/(sin(x))

Problem

d()/d(x)cot(x)/sin(x)

Solution

1. Rewrite the expression using the definition of the cotangent function, cot(x)=cos(x)/sin(x) to simplify the fraction before differentiating.

cot(x)/sin(x)=cos(x)/sin(x)/sin(x)

2. Simplify the complex fraction by multiplying the denominators.

cos(x)/sin2(x)

3. Apply the quotient rule, which states d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2) where u=cos(x) and v=sin2(x)

d()/d(x)cos(x)/sin2(x)=(sin2(x)d(cos(x))/d(x)−cos(x)d(sin2(x))/d(x))/((sin2(x))2)

4. Differentiate the individual components using the power rule and chain rule where necessary.

d(cos(x))/d(x)=−sin(x)

d(sin2(x))/d(x)=2*sin(x)*cos(x)

5. Substitute these derivatives back into the quotient rule formula.

(sin2(x)*(−sin(x))−cos(x)*(2*sin(x)*cos(x)))/sin4(x)

6. Factor out sin(x) from the numerator to simplify the expression.

(sin(x)*(−sin2(x)−2*cos2(x)))/sin4(x)

7. Cancel the common factor of sin(x) and use the Pythagorean identity sin2(x)=1−cos2(x) to further simplify.

(−sin2(x)−2*cos2(x))/sin3(x)

(−(1−cos2(x))−2*cos2(x))/sin3(x)

(−1−cos2(x))/sin3(x)

Final Answer

d()/d(x)cot(x)/sin(x)=−(1+cos2(x))/sin3(x)

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