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

Problem

d()/d(x)(x2−1)/(x2+x+1)

Solution

1. Identify the rule needed for the derivative of a quotient, which is the quotient rule: d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

2. Assign the numerator and denominator functions: u=x2−1 and v=x2+x+1

3. Differentiate the individual components: d(x2−1)/d(x)=2*x and d(x2+x+1)/d(x)=2*x+1

4. Substitute these into the quotient rule formula:

((x2+x+1)*(2*x)−(x2−1)*(2*x+1))/((x2+x+1)2)

5. Expand the terms in the numerator:

((2*x3+2*x2+2*x)−(2*x3+x2−2*x−1))/((x2+x+1)2)

6. Simplify the numerator by combining like terms:

(2*x3+2*x2+2*x−2*x3−x2+2*x+1)/((x2+x+1)2)

(x2+4*x+1)/((x2+x+1)2)

Final Answer

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

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