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

Problem

d2()/(d(x)2)(x2−1)/(x2+1)

Solution

1. Apply the quotient rule to find the first derivative ƒ(x)′ The quotient rule is d()/d(x)u/v=(vd(u)/d(x)−ud(v)/d(x))/(v2)

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

2. Simplify the numerator of the first derivative by distributing and combining like terms.

ƒ(x)′=(2*x3+2*x−2*x3+2*x)/((x2+1)2)

ƒ(x)′=(4*x)/((x2+1)2)

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

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

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

(d(4)*x)/d(x)=4

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

5. Substitute the components back into the expression for ƒ(x)″

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

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

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

ƒ(x)″=(4*(x2+1−4*x2))/((x2+1)3)

7. Combine like terms inside the brackets to reach the final simplified form.

ƒ(x)″=(4*(1−3*x2))/((x2+1)3)

Final Answer

d2()/(d(x)2)(x2−1)/(x2+1)=(4−12*x2)/((x2+1)3)

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