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

Problem

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

Solution

1. Express the cotangent function in terms of sine and cosine using the quotient identity.

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

2. Apply the quotient rule for differentiation, which states that for a function u/v the derivative is (vd(u)/d(x)−ud(v)/d(x))/(v2)

d(cot(x))/d(x)=(sin(x)d(cos(x))/d(x)−cos(x)d(sin(x))/d(x))/sin2(x)

3. Differentiate the numerator terms using the basic rules d(cos(x))/d(x)=−sin(x) and d(sin(x))/d(x)=cos(x)

d(cot(x))/d(x)=(sin(x)*(−sin(x))−cos(x)*(cos(x)))/sin2(x)

4. Simplify the numerator by distributing and factoring out a negative sign.

d(cot(x))/d(x)=(−sin2(x)−cos2(x))/sin2(x)

d(cot(x))/d(x)=(−(sin2(x)+cos2(x)))/sin2(x)

5. Apply the Pythagorean identity sin2(x)+cos2(x)=1 to the numerator.

d(cot(x))/d(x)=(−1)/sin2(x)

6. Use the reciprocal identity csc(x)=1/sin(x) to write the final result.

d(cot(x))/d(x)=−csc2(x)

Final Answer

d(cot(x))/d(x)=−csc2(x)

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