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

Problem

d()/d(x)(1+sin(x))/(1−sin(x))

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction of two functions, apply 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+sin(x) and v=1−sin(x)

3. Differentiate the individual components.

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

d(1−sin(x))/d(x)=−cos(x)

4. Substitute these derivatives into the quotient rule formula.

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

5. Distribute the cos(x) and −cos(x) terms in the numerator.

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

6. Simplify the numerator by combining like terms. The terms −sin(x)*cos(x) and sin(x)*cos(x) cancel out.

(2*cos(x))/((1−sin(x))2)

Final Answer

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

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