Find the 2nd Derivative f(x)=x/(x^2+1)

Problem

ƒ(x)=x/(x^2+1)

Solution

1. Identify the function ƒ(x)=x/(x^2+1) and prepare to use the quotient rule for the first derivative.

2. Apply the quotient rule to find ƒ(x)^′ where the rule is d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v^2)

ƒ(x)^′=((x^2+1)*(1)−(x)*(2*x))/((x^2+1)^2)

3. Simplify the numerator of the first derivative.

ƒ(x)^′=(x^2+1−2*x^2)/((x^2+1)^2)

ƒ(x)^′=(1−x^2)/((x^2+1)^2)

4. Apply the quotient rule again to find the second derivative ƒ(x)^″

ƒ(x)^″=((x^2+1)^2d(1−x^2)/d(x)−(1−x^2)d(x^2+1)/d(x))/(((x^2+1)^2)^2)

5. Differentiate the components using the power rule and the chain rule.

ƒ(x)^″=((x^2+1)^2*(−2*x)−(1−x^2)*(2*(x^2+1)*(2*x)))/((x^2+1)^4)

6. Factor out the common term (x^2+1) from the numerator to simplify.

ƒ(x)^″=((x^2+1)*[(x^2+1)*(−2*x)−(1−x^2)*(4*x)])/((x^2+1)^4)

ƒ(x)^″=(−2*x^3−2*x−(4*x−4*x^3))/((x^2+1)^3)

7. Combine like terms in the numerator.

ƒ(x)^″=(−2*x^3−2*x−4*x+4*x^3)/((x^2+1)^3)

ƒ(x)^″=(2*x^3−6*x)/((x^2+1)^3)

8. Factor the final expression for the second derivative.

ƒ(x)^″=(2*x*(x^2−3))/((x^2+1)^3)

Final Answer

ƒ(x)^″=(2*x*(x^2−3))/((x^2+1)^3)

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