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

Problem

d^2()/(d(x)^2)(x^2−1)/(x^2+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))/(v^2)

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

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

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

ƒ(x)^′=(4*x)/((x^2+1)^2)

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

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

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

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

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

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

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

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

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

ƒ(x)^″=(4*(x^2+1−4*x^2))/((x^2+1)^3)

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

ƒ(x)^″=(4*(1−3*x^2))/((x^2+1)^3)

Final Answer

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

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