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

Problem

ƒ(x)=x/(x2+1)

Solution

1. Identify the function ƒ(x)=x/(x2+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))/(v2)

ƒ(x)′=((x2+1)*(1)−(x)*(2*x))/((x2+1)2)

3. Simplify the numerator of the first derivative.

ƒ(x)′=(x2+1−2*x2)/((x2+1)2)

ƒ(x)′=(1−x2)/((x2+1)2)

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

ƒ(x)″=((x2+1)2d(1−x2)/d(x)−(1−x2)d(x2+1)/d(x))/(((x2+1)2)2)

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

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

6. Factor out the common term (x2+1) from the numerator to simplify.

ƒ(x)″=((x2+1)*[(x2+1)*(−2*x)−(1−x2)*(4*x)])/((x2+1)4)

ƒ(x)″=(−2*x3−2*x−(4*x−4*x3))/((x2+1)3)

7. Combine like terms in the numerator.

ƒ(x)″=(−2*x3−2*x−4*x+4*x3)/((x2+1)3)

ƒ(x)″=(2*x3−6*x)/((x2+1)3)

8. Factor the final expression for the second derivative.

ƒ(x)″=(2*x*(x2−3))/((x2+1)3)

Final Answer

ƒ(x)″=(2*x*(x2−3))/((x2+1)3)

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