Find the Derivative - d/dx y=(x^2-1)/(x^2+x+1)

Problem

d()/d(x)(x2−1)/(x2+x+1)

Solution

1. Identify the rule needed for differentiation. Since the function is a quotient of two expressions, use the quotient rule:

d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator to variables. Let:

u=x2−1

v=x2+x+1

3. Differentiate both parts with respect to x

d(u)/d(x)=2*x

d(v)/d(x)=2*x+1

4. Substitute these values into the quotient rule formula:

d(y)/d(x)=((x2+x+1)*(2*x)−(x2−1)*(2*x+1))/((x2+x+1)2)

5. Expand the terms in the numerator:

(x2+x+1)*(2*x)=2*x3+2*x2+2*x

(x2−1)*(2*x+1)=2*x3+x2−2*x−1

6. Subtract the expanded terms to simplify the numerator:

(2*x3+2*x2+2*x)−(2*x3+x2−2*x−1)

2*x3−2*x3+2*x2−x2+2*x+2*x+1

x2+4*x+1

Final Answer

d()/d(x)(x2−1)/(x2+x+1)=(x2+4*x+1)/((x2+x+1)2)

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