Find the 2nd Derivative f(x)=5/(x^2+5)

Problem

ƒ(x)=5/(x2+5)

Solution

1. Rewrite the function using a negative exponent to prepare for the chain rule.

ƒ(x)=5*(x2+5)(−1)

2. Find the first derivative by applying the power rule and the chain rule.

ƒ(x)′=−5*(x2+5)(−2)⋅d(x2+5)/d(x)

ƒ(x)′=−5*(x2+5)(−2)⋅2*x

ƒ(x)′=−10*x*(x2+5)(−2)

3. Find the second derivative by applying the product rule and the chain rule to the first derivative.

ƒ(x)″=(d(−)*10*x)/d(x)⋅(x2+5)(−2)+(−10*x)⋅d(x2+5)/d(x)

ƒ(x)″=−10*(x2+5)(−2)+(−10*x)⋅(−2*(x2+5)(−3)⋅2*x)

4. Simplify the expression by performing the multiplication in the second term.

ƒ(x)″=−10*(x2+5)(−2)+40*x2*(x2+5)(−3)

5. Factor out the common term 10*(x2+5)(−3) to combine the terms into a single fraction.

ƒ(x)″=10*(x2+5)(−3)⋅(−(x2+5)+4*x2)

ƒ(x)″=10*(x2+5)(−3)⋅(3*x2−5)

6. Rewrite the final expression in fractional form.

ƒ(x)″=(10*(3*x2−5))/((x2+5)3)

Final Answer

ƒ(x)″=(30*x2−50)/((x2+5)3)

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