Find the 2nd Derivative f(x) = square root of x^2+11

Problem

ƒ(x)=√(,x2+11)

Solution

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

ƒ(x)=(x2+11)(1/2)

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

ƒ(x)′=1/2*(x2+11)(−1/2)⋅d(x2+11)/d(x)

3. Differentiate the inner function and simplify the expression.

ƒ(x)′=1/2*(x2+11)(−1/2)⋅2*x

ƒ(x)′=x*(x2+11)(−1/2)

4. Apply the quotient rule or product rule to find the second derivative, ƒ(x)″ Using the quotient rule on x/√(,x2+11)

ƒ(x)″=(√(,x2+11)⋅d(x)/d(x)−x⋅d(√(,x2+11))/d(x))/((√(,x2+11))2)

5. Substitute the derivative of the square root found in step 3 into the quotient rule formula.

ƒ(x)″=(√(,x2+11)−x⋅x/√(,x2+11))/(x2+11)

6. Simplify the numerator by finding a common denominator.

ƒ(x)″=(x2+11−x2)/√(,x2+11)/(x2+11)

7. Combine the terms into a single fraction.

ƒ(x)″=11/((x2+11)(3/2))

Final Answer

ƒ(x)″=11/((x2+11)√(,x2+11))

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