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

Problem

d()/d(x)sin(x)/(1+cos(x))

Solution

1. Identify the rule needed for the derivative of a quotient, 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=sin(x) and v=1+cos(x)

3. Differentiate the individual components: d(sin(x))/d(x)=cos(x) and d(1+cos(x))/d(x)=−sin(x)

4. Substitute these into the quotient rule formula:

((1+cos(x))*(cos(x))−(sin(x))*(−sin(x)))/((1+cos(x))2)

5. Distribute and simplify the numerator:

(cos(x)+cos2(x)+sin2(x))/((1+cos(x))2)

6. Apply the Pythagorean identity sin2(x)+cos2(x)=1

(cos(x)+1)/((1+cos(x))2)

7. Cancel the common factor (1+cos(x)) from the numerator and denominator:

1/(1+cos(x))

Final Answer

d()/d(x)sin(x)/(1+cos(x))=1/(1+cos(x))

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