Find the 2nd Derivative f(x) = square root of x^2+9

Problem

ƒ(x)=√(,x2+9)

Solution

1. Rewrite the function using a fractional exponent to prepare for differentiation.

ƒ(x)=(x2+9)(1/2)

2. Apply the power rule and chain rule to find the first derivative.

ƒ(x)′=1/2*(x2+9)(−1/2)⋅d(x2+9)/d(x)

3. Differentiate the inner function and simplify the expression.

ƒ(x)′=1/2*(x2+9)(−1/2)⋅2*x

ƒ(x)′=x*(x2+9)(−1/2)

4. Apply the product rule to find the second derivative, where the two functions are x and (x2+9)(−1/2)

ƒ(x)″=d(x)/d(x)⋅(x2+9)(−1/2)+x⋅d(x2+9)/d(x)

5. Differentiate the second term using the chain rule.

ƒ(x)″=1⋅(x2+9)(−1/2)+x⋅(−1/2*(x2+9)(−3/2)⋅2*x)

6. Simplify the terms by multiplying the factors.

ƒ(x)″=(x2+9)(−1/2)−x2*(x2+9)(−3/2)

7. Factor out the common term with the lowest exponent, which is (x2+9)(−3/2)

ƒ(x)″=(x2+9)(−3/2)⋅((x2+9)1−x2)

8. Simplify the expression inside the parentheses.

ƒ(x)″=(x2+9)(−3/2)⋅9

ƒ(x)″=9/((x2+9)(3/2))

Final Answer

d2(√(,x2+9))/(d(x)2)=9/((x2+9)(3/2))

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