Find the Derivative - d/dx (x+y)/(x-y)

Problem

d()/d(x)(x+y)/(x−y)

Solution

1. Identify the rule needed for differentiation. Since the expression is a fraction involving x and y use the quotient rule and assume y is a function of x (implicit differentiation).

2. Apply the quotient rule formula, which is d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2) Let u=x+y and v=x−y

3. Differentiate the numerator and denominator with respect to x

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

d(x−y)/d(x)=1−d(y)/d(x)

4. Substitute these derivatives back into the quotient rule formula.

((x−y)*(1+d(y)/d(x))−(x+y)*(1−d(y)/d(x)))/((x−y)2)

5. Expand the terms in the numerator.

(x+xd(y)/d(x)−y−yd(y)/d(x)−(x−xd(y)/d(x)+y−yd(y)/d(x)))/((x−y)2)

6. Distribute the negative sign and combine like terms.

(x+xd(y)/d(x)−y−yd(y)/d(x)−x+xd(y)/d(x)−y+yd(y)/d(x))/((x−y)2)

7. Simplify the numerator by canceling x and −x and −yd(y)/d(x) and yd(y)/d(x)

(2*xd(y)/d(x)−2*y)/((x−y)2)

Final Answer

d()/d(x)(x+y)/(x−y)=(2*xd(y)/d(x)−2*y)/((x−y)2)

Want more problems? Check here!