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

Problem

d()/d(x)(x2+2*x−2)/(x2−2*x+2)

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 functions. Let u=x2+2*x−2 and v=x2−2*x+2

3. Differentiate u and v with respect to x

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

d(v)/d(x)=2*x−2

4. Substitute these components into the quotient rule formula.

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

5. Expand the products in the numerator.

(x2−2*x+2)*(2*x+2)=2*x3+2*x2−4*x2−4*x+4*x+4=2*x3−2*x2+4

(x2+2*x−2)*(2*x−2)=2*x3−2*x2+4*x2−4*x−4*x+4=2*x3+2*x2−8*x+4

6. Subtract the expanded terms and simplify the numerator.

(2*x3−2*x2+4)−(2*x3+2*x2−8*x+4)=−4*x2+8*x

7. Factor the resulting numerator to reach the final simplified form.

−4*x2+8*x=−4*x*(x−2)

Final Answer

d()/d(x)(x2+2*x−2)/(x2−2*x+2)=(−4*x2+8*x)/((x2−2*x+2)2)

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