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

Problem

ƒ(x)=√(,x^2+11)

Solution

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

ƒ(x)=(x^2+11)^(1/2)

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

ƒ(x)^′=1/2*(x^2+11)^(−1/2)⋅d(x^2+11)/d(x)

3. Differentiate the inner function and simplify the expression.

ƒ(x)^′=1/2*(x^2+11)^(−1/2)⋅2*x

ƒ(x)^′=x*(x^2+11)^(−1/2)

4. Apply the quotient rule or product rule to find the second derivative, ƒ(x)^″ Using the quotient rule on x/√(,x^2+11)

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

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

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

6. Simplify the numerator by finding a common denominator.

ƒ(x)^″=(x^2+11−x^2)/√(,x^2+11)/(x^2+11)

7. Combine the terms into a single fraction.

ƒ(x)^″=11/((x^2+11)^(3/2))

Final Answer

ƒ(x)^″=11/((x^2+11)√(,x^2+11))

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