Find the 2nd Derivative f(x) = square root of x^2+27

Problem

ƒ(x)=√(,x^2+27)

Solution

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

ƒ(x)=(x^2+27)^(1/2)

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

ƒ(x)^′=1/2*(x^2+27)^(−1/2)⋅2*x

3. Simplify the first derivative by canceling the constant factors.

ƒ(x)^′=x*(x^2+27)^(−1/2)

4. Apply the product rule to find the second derivative ƒ(x)^″ where u=x and v=(x^2+27)^(−1/2)

ƒ(x)^″=d(x)/d(x)⋅(x^2+27)^(−1/2)+x⋅d(x^2+27)/d(x)

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

ƒ(x)^″=1⋅(x^2+27)^(−1/2)+x⋅(−1/2*(x^2+27)^(−3/2)⋅2*x)

6. Simplify the expression by combining terms.

ƒ(x)^″=(x^2+27)^(−1/2)−x^2*(x^2+27)^(−3/2)

7. Factor out the common term (x^2+27)^(−3/2) to simplify the fraction.

ƒ(x)^″=(x^2+27)^(−3/2)⋅((x^2+27)^1−x^2)

8. Combine the remaining terms in the numerator.

ƒ(x)^″=27/((x^2+27)^(3/2))

Final Answer

ƒ(x)^″=27/((x^2+27)√(,x^2+27))

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