find the second derivative of f(x)=3/(x^2+12)

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second derivative 3/(x^2+12)

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write it as

y = 3(x^2 + 12)^-1
y' = -3(x^2 + 12)^-2 (2x)
= -6x(x^2 + 12)^-2

y'' = (-6x)(-2)(x^2 + 12)^-3 (2x) - 6(x^2 + 12)^-2
= 24x^2 (x^2+12)^-3 - 6(x^2+12)^-2
= 6(x^2+12)^-3 [ 4x^2 - x^2 - 12]
= 6(x^2+12)^-3 [ 3x^2 -12]
= 18(x^2 -4)/(x^2 + 12)^3
or
18(x^2-4)/(x^2 + 12)^3

In the second derivative, second line, what happened to the -2 power on the end of the line. It looks like you just dropped it. I can't see where it was factored out.

To find the second derivative of the function f(x) = 3/(x^2 + 12), we need to follow a few steps.

Step 1: Determine the first derivative of f(x).
Step 2: Take the derivative of the first derivative obtained in Step 1 to find the second derivative.

Let's start with Step 1:

The given function is f(x) = 3/(x^2 + 12).

Step 1: Determine the first derivative of f(x).

To find the first derivative, we need to apply the quotient rule:

f'(x) = [(3)(d/dx(x^2 + 12)) - (x^2 + 12)(d/dx(3))]/(x^2 + 12)^2.

Simplifying this expression:

f'(x) = [3(2x) - 0]/(x^2 + 12)^2
= 6x/(x^2 + 12)^2.

Now we move on to Step 2 to find the second derivative:

Step 2: Take the derivative of the first derivative obtained in Step 1 to find the second derivative.

To find the second derivative, we take the derivative of f'(x):

f''(x) = d/dx(6x/(x^2 + 12)^2).

To simplify this expression, we can use the quotient rule again:

f''(x) = [(d/dx(6x))(x^2 + 12)^2 - (6x)(d/dx(x^2 + 12)^2)]/(x^2 + 12)^4.

Let's calculate the derivatives in this expression:

d/dx(6x) = 6,
d/dx(x^2 + 12)^2 = 2(x^2 + 12)(d/dx(x^2 + 12)) = 2(x^2 + 12)(2x) = 4x(x^2 + 12),

Substituting these derivatives back into the expression for f''(x):

f''(x) = [(6)(x^2 + 12)^2 - (6x)(4x(x^2 + 12)))/(x^2 + 12)^4
= (6(x^2 + 12)^2 - 24x^2(x^2 + 12))/(x^2 + 12)^4.

Simplifying further:

f''(x) = (6(x^2 + 12)^2 - 24x^4 - 288x^2)/(x^2 + 12)^4.

Therefore, the second derivative of f(x) = 3/(x^2 + 12) is f''(x) = (6(x^2 + 12)^2 - 24x^4 - 288x^2)/(x^2 + 12)^4.