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

Problem

ƒ(x)=√(,x2+27)

Solution

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

ƒ(x)=(x2+27)(1/2)

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

ƒ(x)′=1/2*(x2+27)(−1/2)⋅2*x

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

ƒ(x)′=x*(x2+27)(−1/2)

4. Apply the product rule to find the second derivative ƒ(x)″ where u=x and v=(x2+27)(−1/2)

ƒ(x)″=d(x)/d(x)⋅(x2+27)(−1/2)+x⋅d(x2+27)/d(x)

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

ƒ(x)″=1⋅(x2+27)(−1/2)+x⋅(−1/2*(x2+27)(−3/2)⋅2*x)

6. Simplify the expression by combining terms.

ƒ(x)″=(x2+27)(−1/2)−x2*(x2+27)(−3/2)

7. Factor out the common term (x2+27)(−3/2) to simplify the fraction.

ƒ(x)″=(x2+27)(−3/2)⋅((x2+27)1−x2)

8. Combine the remaining terms in the numerator.

ƒ(x)″=27/((x2+27)(3/2))

Final Answer

ƒ(x)″=27/((x2+27)√(,x2+27))

Want more problems? Check here!