Find dy/dx y=(1-cos(x))/(sin(x))

Problem

d()/d(x)(1−cos(x))/sin(x)

Solution

1. Identify the function as a quotient and apply the quotient rule, which states that for y=u/v the derivative is d(y)/d(x)=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Differentiate the numerator u=1−cos(x) to get d(u)/d(x)=sin(x)

3. Differentiate the denominator v=sin(x) to get d(v)/d(x)=cos(x)

4. Substitute these derivatives into the quotient rule formula.

d(y)/d(x)=(sin(x)*sin(x)−(1−cos(x))*cos(x))/sin2(x)

5. Distribute the cos(x) in the numerator.

d(y)/d(x)=(sin2(x)−(cos(x)−cos2(x)))/sin2(x)

6. Simplify the numerator by removing parentheses and grouping terms.

d(y)/d(x)=(sin2(x)+cos2(x)−cos(x))/sin2(x)

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

d(y)/d(x)=(1−cos(x))/sin2(x)

8. Use the identity sin2(x)=1−cos2(x) to further simplify the expression.

d(y)/d(x)=(1−cos(x))/(1−cos2(x))

9. Factor the denominator as a difference of squares.

d(y)/d(x)=(1−cos(x))/((1−cos(x))*(1+cos(x)))

10. Cancel the common factor (1−cos(x)) from the numerator and denominator.

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

Final Answer

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

Want more problems? Check here!