Find the 2nd Derivative f(x) = square root of x^2+12

Problem

ƒ(x)=√(,x2+12)

Solution

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

ƒ(x)=(x2+12)(1/2)

2. Apply the power rule and chain rule to find the first derivative ƒ(x)′

ƒ(x)′=1/2*(x2+12)(−1/2)⋅d(x2+12)/d(x)

3. Differentiate the inner function and simplify the expression.

ƒ(x)′=1/2*(x2+12)(−1/2)⋅2*x

ƒ(x)′=x*(x2+12)(−1/2)

4. Apply the product rule to find the second derivative ƒ(x)″ where u=x and v=(x2+12)(−1/2)

ƒ(x)″=d(x)/d(x)⋅(x2+12)(−1/2)+x⋅d(x2+12)/d(x)

5. Differentiate the second term using the chain rule.

ƒ(x)″=1⋅(x2+12)(−1/2)+x⋅(−1/2*(x2+12)(−3/2)⋅2*x)

6. Simplify the expression by combining terms.

ƒ(x)″=(x2+12)(−1/2)−x2*(x2+12)(−3/2)

7. Factor out the common term (x2+12)(−3/2) to find a common denominator.

ƒ(x)″=(x2+12)(−3/2)⋅((x2+12)1−x2)

ƒ(x)″=(x2+12)(−3/2)⋅12

8. Rewrite the final result in radical form.

ƒ(x)″=12/((x2+12)(3/2))

Final Answer

d2(√(,x2+12))/(d(x)2)=12/((x2+12)√(,x2+12))

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