Find the Derivative - d/dx ((sin(x))/(1+cos(x)))^2

Problem

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

Solution

1. Apply the power rule and the chain rule to the outer exponent.

d()/d(x)*(sin(x)/(1+cos(x)))2=2*(sin(x)/(1+cos(x)))⋅d()/d(x)sin(x)/(1+cos(x))

2. Apply the quotient rule to find the derivative of the inner fraction.

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

3. Simplify the numerator using the Pythagorean identity sin2(x)+cos2(x)=1

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

4. Reduce the fraction by canceling the common factor (1+cos(x))

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

5. Substitute back into the chain rule expression.

2*(sin(x)/(1+cos(x)))⋅1/(1+cos(x))=(2*sin(x))/((1+cos(x))2)

Final Answer

d()/d(x)*(sin(x)/(1+cos(x)))2=(2*sin(x))/((1+cos(x))2)

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