Calculus Examples

$\frac{4 x + 4}{x^{2} + 3}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 4 x + 4$ and $g (x) = x^{2} + 3$ .

$f' (x) = \frac{(x^{2} + 3) \frac{d}{d x} (4 x + 4) - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $4 x + 4$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [4]$ .

$f' (x) = \frac{(x^{2} + 3) (\frac{d}{d x} (4 x) + \frac{d}{d x} (4)) - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$f' (x) = \frac{(x^{2} + 3) (4 \frac{d}{d x} (x) + \frac{d}{d x} (4)) - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f' (x) = \frac{(x^{2} + 3) (4 \cdot 1 + \frac{d}{d x} (4)) - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2.4

Multiply $4$ by $1$ .

$f' (x) = \frac{(x^{2} + 3) (4 + \frac{d}{d x} (4)) - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2.5

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$f' (x) = \frac{(x^{2} + 3) (4 + 0) - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $4$ and $0$ .

$f' (x) = \frac{(x^{2} + 3) \cdot 4 - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2.6.2

Move $4$ to the left of $x^{2} + 3$ .

$f' (x) = \frac{4 \cdot (x^{2} + 3) - (4 x + 4) \frac{d}{d x} (x^{2} + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.2.7

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

$f' (x) = \frac{4 (x^{2} + 3) - (4 x + 4) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (3))}{{(x^{2} + 3)}^{2}}$

Step 1.2.8

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f' (x) = \frac{4 (x^{2} + 3) - (4 x + 4) (2 x + \frac{d}{d x} (3))}{{(x^{2} + 3)}^{2}}$

Step 1.2.9

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f' (x) = \frac{4 (x^{2} + 3) - (4 x + 4) (2 x + 0)}{{(x^{2} + 3)}^{2}}$

Step 1.2.10

Simplify the expression.

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Step 1.2.10.1

Add $2 x$ and $0$ .

$f' (x) = \frac{4 (x^{2} + 3) - (4 x + 4) (2 x)}{{(x^{2} + 3)}^{2}}$

Step 1.2.10.2

Multiply $2$ by $- 1$ .

$f' (x) = \frac{4 (x^{2} + 3) - 2 (4 x + 4) x}{{(x^{2} + 3)}^{2}}$

Step 1.3

Simplify.

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Step 1.3.1

Apply the distributive property.

$f' (x) = \frac{4 x^{2} + 4 \cdot 3 - 2 (4 x + 4) x}{{(x^{2} + 3)}^{2}}$

Step 1.3.2

Apply the distributive property.

$f' (x) = \frac{4 x^{2} + 4 \cdot 3 + (- 2 (4 x) - 2 \cdot 4) x}{{(x^{2} + 3)}^{2}}$

Step 1.3.3

Apply the distributive property.

$f' (x) = \frac{4 x^{2} + 4 \cdot 3 - 2 (4 x) x - 2 \cdot (4 x)}{{(x^{2} + 3)}^{2}}$

Step 1.3.4

Simplify the numerator.

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Step 1.3.4.1

Simplify each term.

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Step 1.3.4.1.1

Multiply $4$ by $3$ .

$f' (x) = \frac{4 x^{2} + 12 - 2 (4 x) x - 2 \cdot (4 x)}{{(x^{2} + 3)}^{2}}$

Step 1.3.4.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.4.1.2.1

Move $x$ .

$f' (x) = \frac{4 x^{2} + 12 - 2 (4 (x \cdot x)) - 2 \cdot (4 x)}{{(x^{2} + 3)}^{2}}$

Step 1.3.4.1.2.2

Multiply $x$ by $x$ .

$f' (x) = \frac{4 x^{2} + 12 - 2 (4 x^{2}) - 2 \cdot (4 x)}{{(x^{2} + 3)}^{2}}$

Step 1.3.4.1.3

Multiply $4$ by $- 2$ .

$f' (x) = \frac{4 x^{2} + 12 - 8 x^{2} - 2 \cdot (4 x)}{{(x^{2} + 3)}^{2}}$

Step 1.3.4.1.4

Multiply $- 2$ by $4$ .

$f' (x) = \frac{4 x^{2} + 12 - 8 x^{2} - 8 x}{{(x^{2} + 3)}^{2}}$

Step 1.3.4.2

Subtract $8 x^{2}$ from $4 x^{2}$ .

$f' (x) = \frac{- 4 x^{2} + 12 - 8 x}{{(x^{2} + 3)}^{2}}$

Step 1.3.5

Reorder terms.

$f' (x) = \frac{- 4 x^{2} - 8 x + 12}{{(x^{2} + 3)}^{2}}$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Factor $4$ out of $- 4 x^{2} - 8 x + 12$ .

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Step 1.3.6.1.1

Factor $4$ out of $- 4 x^{2}$ .

$f' (x) = \frac{4 (- x^{2}) - 8 x + 12}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.1.2

Factor $4$ out of $- 8 x$ .

$f' (x) = \frac{4 (- x^{2}) + 4 (- 2 x) + 12}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.1.3

Factor $4$ out of $12$ .

$f' (x) = \frac{4 (- x^{2}) + 4 (- 2 x) + 4 (3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.1.4

Factor $4$ out of $4 (- x^{2}) + 4 (- 2 x)$ .

$f' (x) = \frac{4 (- x^{2} - 2 x) + 4 (3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.1.5

Factor $4$ out of $4 (- x^{2} - 2 x) + 4 (3)$ .

$f' (x) = \frac{4 (- x^{2} - 2 x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.2

Factor by grouping.

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Step 1.3.6.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot 3 = - 3$ and whose sum is $b = - 2$ .

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Step 1.3.6.2.1.1

Factor $- 2$ out of $- 2 x$ .

$f' (x) = \frac{4 (- x^{2} - 2 x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.2.1.2

Rewrite $- 2$ as $1$ plus $- 3$

$f' (x) = \frac{4 (- x^{2} + (1 - 3) x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.2.1.3

Apply the distributive property.

$f' (x) = \frac{4 (- x^{2} + 1 x - 3 x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.2.1.4

Multiply $x$ by $1$ .

$f' (x) = \frac{4 (- x^{2} + x - 3 x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.2.2

Factor out the greatest common factor from each group.

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Step 1.3.6.2.2.1

Group the first two terms and the last two terms.

$f' (x) = \frac{4 ((- x^{2} + x) - 3 x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.2.2.2

Factor out the greatest common factor (GCF) from each group.

$f' (x) = \frac{4 (x (- x + 1) + 3 (- x + 1))}{{(x^{2} + 3)}^{2}}$

Step 1.3.6.2.3

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$f' (x) = \frac{4 ((- x + 1) (x + 3))}{{(x^{2} + 3)}^{2}}$

$f' (x) = \frac{4 (- x + 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.7

Factor $- 1$ out of $- x$ .

$f' (x) = \frac{4 (- (x) + 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.8

Rewrite $1$ as $- 1 (- 1)$ .

$f' (x) = \frac{4 (- (x) - 1 \cdot - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.9

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$f' (x) = \frac{4 (- (x - 1)) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.10

Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$f' (x) = \frac{4 (- 1 (x - 1)) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.11

Move the negative in front of the fraction.

$f' (x) = - \frac{(4 (x - 1)) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 1.3.12

Reorder factors in $- \frac{(4 (x - 1)) (x + 3)}{{(x^{2} + 3)}^{2}}$ .

$f' (x) = - \frac{4 (x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 2

Find the second derivative.

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Step 2.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- \frac{4 (x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$ with respect to $x$ is $- 4 \frac{d}{d x} [\frac{(x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}]$ .

$f'' (x) = - 4 \frac{d}{d x} \frac{(x - 1) (x + 3)}{{(x^{2} + 3)}^{2}}$

Step 2.2

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} \frac{d}{d x} ((x - 1) (x + 3)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{({(x^{2} + 3)}^{2})}^{2}}$

Step 2.3

Multiply the exponents in ${({(x^{2} + 3)}^{2})}^{2}$ .

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Step 2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} \frac{d}{d x} ((x - 1) (x + 3)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{2 \cdot 2}}$

Step 2.3.2

Multiply $2$ by $2$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} \frac{d}{d x} ((x - 1) (x + 3)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.4

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x - 1$ and $g (x) = x + 3$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} ((x - 1) \frac{d}{d x} (x + 3) + (x + 3) \frac{d}{d x} (x - 1)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5

Differentiate.

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Step 2.5.1

By the Sum Rule, the derivative of $x + 3$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} ((x - 1) (\frac{d}{d x} (x) + \frac{d}{d x} (3)) + (x + 3) \frac{d}{d x} (x - 1)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} ((x - 1) (1 + \frac{d}{d x} (3)) + (x + 3) \frac{d}{d x} (x - 1)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} ((x - 1) (1 + 0) + (x + 3) \frac{d}{d x} (x - 1)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.4

Simplify the expression.

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Step 2.5.4.1

Add $1$ and $0$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} ((x - 1) \cdot 1 + (x + 3) \frac{d}{d x} (x - 1)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.4.2

Multiply $x - 1$ by $1$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (x - 1 + (x + 3) \frac{d}{d x} (x - 1)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.5

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (x - 1 + (x + 3) (\frac{d}{d x} (x) + \frac{d}{d x} (- 1))) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.6

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (x - 1 + (x + 3) (1 + \frac{d}{d x} (- 1))) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.7

Since $- 1$ is constant with respect to $x$ , the derivative of $- 1$ with respect to $x$ is $0$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (x - 1 + (x + 3) (1 + 0)) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.8

Simplify by adding terms.

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Step 2.5.8.1

Add $1$ and $0$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (x - 1 + (x + 3) \cdot 1) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.8.2

Multiply $x + 3$ by $1$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (x - 1 + x + 3) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.8.3

Add $x$ and $x$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (2 x - 1 + 3) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.5.8.4

Add $- 1$ and $3$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (2 x + 2) - (x - 1) (x + 3) \frac{d}{d x} {(x^{2} + 3)}^{2}}{{(x^{2} + 3)}^{4}}$

Step 2.6

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} + 3$ .

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Step 2.6.1

To apply the Chain Rule, set $u$ as $x^{2} + 3$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (2 x + 2) - (x - 1) (x + 3) (\frac{d}{d u} (u^{2}) \frac{d}{d x} (x^{2} + 3))}{{(x^{2} + 3)}^{4}}$

Step 2.6.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 2$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (2 x + 2) - (x - 1) (x + 3) (2 u \frac{d}{d x} (x^{2} + 3))}{{(x^{2} + 3)}^{4}}$

Step 2.6.3

Replace all occurrences of $u$ with $x^{2} + 3$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (2 x + 2) - (x - 1) (x + 3) (2 (x^{2} + 3) \frac{d}{d x} (x^{2} + 3))}{{(x^{2} + 3)}^{4}}$

Step 2.7

Simplify with factoring out.

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Step 2.7.1

Multiply $2$ by $- 1$ .

$f'' (x) = - 4 \frac{{(x^{2} + 3)}^{2} (2 x + 2) - 2 (x - 1) (x + 3) ((x^{2} + 3) \frac{d}{d x} (x^{2} + 3))}{{(x^{2} + 3)}^{4}}$

Step 2.7.2

Factor $x^{2} + 3$ out of ${(x^{2} + 3)}^{2} (2 x + 2) - 2 (x - 1) (x + 3) ((x^{2} + 3) \frac{d}{d x} [x^{2} + 3])$ .

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Step 2.7.2.1

Factor $x^{2} + 3$ out of ${(x^{2} + 3)}^{2} (2 x + 2)$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) ((x^{2} + 3) (2 x + 2)) - 2 (x - 1) (x + 3) ((x^{2} + 3) \frac{d}{d x} (x^{2} + 3))}{{(x^{2} + 3)}^{4}}$

Step 2.7.2.2

Factor $x^{2} + 3$ out of $- 2 (x - 1) (x + 3) ((x^{2} + 3) \frac{d}{d x} [x^{2} + 3])$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) ((x^{2} + 3) (2 x + 2)) + (x^{2} + 3) (- 2 (x - 1) (x + 3) (\frac{d}{d x} (x^{2} + 3)))}{{(x^{2} + 3)}^{4}}$

Step 2.7.2.3

Factor $x^{2} + 3$ out of $(x^{2} + 3) ((x^{2} + 3) (2 x + 2)) + (x^{2} + 3) (- 2 (x - 1) (x + 3) (\frac{d}{d x} [x^{2} + 3]))$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) ((x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (\frac{d}{d x} (x^{2} + 3)))}{{(x^{2} + 3)}^{4}}$

Step 2.8

Cancel the common factors.

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Step 2.8.1

Factor $x^{2} + 3$ out of ${(x^{2} + 3)}^{4}$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) ((x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (\frac{d}{d x} (x^{2} + 3)))}{(x^{2} + 3) {(x^{2} + 3)}^{3}}$

Step 2.8.2

Cancel the common factor.

$f'' (x) = - 4 \frac{(x^{2} + 3) ((x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (\frac{d}{d x} (x^{2} + 3)))}{(x^{2} + 3) {(x^{2} + 3)}^{3}}$

Step 2.8.3

Rewrite the expression.

$f'' (x) = - 4 \frac{(x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (\frac{d}{d x} (x^{2} + 3))}{{(x^{2} + 3)}^{3}}$

Step 2.9

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (\frac{d}{d x} (x^{2}) + \frac{d}{d x} (3))}{{(x^{2} + 3)}^{3}}$

Step 2.10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (2 x + \frac{d}{d x} (3))}{{(x^{2} + 3)}^{3}}$

Step 2.11

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (2 x + 0)}{{(x^{2} + 3)}^{3}}$

Step 2.12

Combine fractions.

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Step 2.12.1

Add $2 x$ and $0$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) (2 x + 2) - 2 (x - 1) (x + 3) (2 x)}{{(x^{2} + 3)}^{3}}$

Step 2.12.2

Multiply $2$ by $- 2$ .

$f'' (x) = - 4 \frac{(x^{2} + 3) (2 x + 2) - 4 (x - 1) (x + 3) x}{{(x^{2} + 3)}^{3}}$

Step 2.12.3

Combine $- 4$ and $\frac{(x^{2} + 3) (2 x + 2) - 4 (x - 1) (x + 3) x}{{(x^{2} + 3)}^{3}}$ .

$f'' (x) = \frac{- 4 ((x^{2} + 3) (2 x + 2) - 4 (x - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.12.4

Move the negative in front of the fraction.

$f'' (x) = - \frac{(4) ((x^{2} + 3) (2 x + 2) - 4 (x - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13

Simplify.

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Step 2.13.1

Apply the distributive property.

$f'' (x) = - \frac{4 ((x^{2} + 3) (2 x + 2) + (- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.2

Apply the distributive property.

$f'' (x) = - \frac{4 ((x^{2} + 3) (2 x + 2)) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3

Simplify the numerator.

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Step 2.13.3.1

Simplify each term.

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Step 2.13.3.1.1

Expand $(x^{2} + 3) (2 x + 2)$ using the FOIL Method.

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Step 2.13.3.1.1.1

Apply the distributive property.

$f'' (x) = - \frac{4 (x^{2} (2 x + 2) + 3 (2 x + 2)) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.1.2

Apply the distributive property.

$f'' (x) = - \frac{4 (x^{2} (2 x) + x^{2} \cdot 2 + 3 (2 x + 2)) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.1.3

Apply the distributive property.

$f'' (x) = - \frac{4 (x^{2} (2 x) + x^{2} \cdot 2 + 3 (2 x) + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2

Simplify each term.

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Step 2.13.3.1.2.1

Rewrite using the commutative property of multiplication.

$f'' (x) = - \frac{4 (2 x^{2} x + x^{2} \cdot 2 + 3 (2 x) + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.13.3.1.2.2.1

Move $x$ .

$f'' (x) = - \frac{4 (2 (x \cdot x^{2}) + x^{2} \cdot 2 + 3 (2 x) + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2.2.2

Multiply $x$ by $x^{2}$ .

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Step 2.13.3.1.2.2.2.1

Raise $x$ to the power of $1$ .

$f'' (x) = - \frac{4 (2 (x \cdot x^{2}) + x^{2} \cdot 2 + 3 (2 x) + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{4 (2 x^{1 + 2} + x^{2} \cdot 2 + 3 (2 x) + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2.2.3

Add $1$ and $2$ .

$f'' (x) = - \frac{4 (2 x^{3} + x^{2} \cdot 2 + 3 (2 x) + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2.3

Move $2$ to the left of $x^{2}$ .

$f'' (x) = - \frac{4 (2 x^{3} + 2 \cdot x^{2} + 3 (2 x) + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2.4

Multiply $2$ by $3$ .

$f'' (x) = - \frac{4 (2 x^{3} + 2 x^{2} + 6 x + 3 \cdot 2) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.2.5

Multiply $3$ by $2$ .

$f'' (x) = - \frac{4 (2 x^{3} + 2 x^{2} + 6 x + 6) + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.3

Apply the distributive property.

$f'' (x) = - \frac{4 (2 x^{3}) + 4 (2 x^{2}) + 4 (6 x) + 4 \cdot 6 + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.4

Simplify.

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Step 2.13.3.1.4.1

Multiply $2$ by $4$ .

$f'' (x) = - \frac{8 x^{3} + 4 (2 x^{2}) + 4 (6 x) + 4 \cdot 6 + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.4.2

Multiply $2$ by $4$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 4 (6 x) + 4 \cdot 6 + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.4.3

Multiply $6$ by $4$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 4 \cdot 6 + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.4.4

Multiply $4$ by $6$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x - 4 \cdot - 1) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.5

Multiply $- 4$ by $- 1$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x + 4) (x + 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.6

Expand $(- 4 x + 4) (x + 3)$ using the FOIL Method.

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Step 2.13.3.1.6.1

Apply the distributive property.

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x (x + 3) + 4 (x + 3)) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.6.2

Apply the distributive property.

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x \cdot x - 4 x \cdot 3 + 4 (x + 3)) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.6.3

Apply the distributive property.

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x \cdot x - 4 x \cdot 3 + 4 x + 4 \cdot 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.7

Simplify and combine like terms.

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Step 2.13.3.1.7.1

Simplify each term.

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Step 2.13.3.1.7.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.13.3.1.7.1.1.1

Move $x$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 (x \cdot x) - 4 x \cdot 3 + 4 x + 4 \cdot 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.7.1.1.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x^{2} - 4 x \cdot 3 + 4 x + 4 \cdot 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.7.1.2

Multiply $3$ by $- 4$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x^{2} - 12 x + 4 x + 4 \cdot 3) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.7.1.3

Multiply $4$ by $3$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x^{2} - 12 x + 4 x + 12) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.7.2

Add $- 12 x$ and $4 x$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 ((- 4 x^{2} - 8 x + 12) x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.8

Apply the distributive property.

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 x^{2} x - 8 x \cdot x + 12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.9

Simplify.

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Step 2.13.3.1.9.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.13.3.1.9.1.1

Move $x$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 (x \cdot x^{2}) - 8 x \cdot x + 12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.9.1.2

Multiply $x$ by $x^{2}$ .

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Step 2.13.3.1.9.1.2.1

Raise $x$ to the power of $1$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 (x \cdot x^{2}) - 8 x \cdot x + 12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.9.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 x^{1 + 2} - 8 x \cdot x + 12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.9.1.3

Add $1$ and $2$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 x^{3} - 8 x \cdot x + 12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.9.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.13.3.1.9.2.1

Move $x$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 x^{3} - 8 (x \cdot x) + 12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.9.2.2

Multiply $x$ by $x$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 x^{3} - 8 x^{2} + 12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.10

Apply the distributive property.

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 + 4 (- 4 x^{3}) + 4 (- 8 x^{2}) + 4 (12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.11

Simplify.

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Step 2.13.3.1.11.1

Multiply $- 4$ by $4$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 - 16 x^{3} + 4 (- 8 x^{2}) + 4 (12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.11.2

Multiply $- 8$ by $4$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 - 16 x^{3} - 32 x^{2} + 4 (12 x)}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.1.11.3

Multiply $12$ by $4$ .

$f'' (x) = - \frac{8 x^{3} + 8 x^{2} + 24 x + 24 - 16 x^{3} - 32 x^{2} + 48 x}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.2

Subtract $16 x^{3}$ from $8 x^{3}$ .

$f'' (x) = - \frac{- 8 x^{3} + 8 x^{2} + 24 x + 24 - 32 x^{2} + 48 x}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.3

Subtract $32 x^{2}$ from $8 x^{2}$ .

$f'' (x) = - \frac{- 8 x^{3} - 24 x^{2} + 24 x + 24 + 48 x}{{(x^{2} + 3)}^{3}}$

Step 2.13.3.4

Add $24 x$ and $48 x$ .

$f'' (x) = - \frac{- 8 x^{3} - 24 x^{2} + 72 x + 24}{{(x^{2} + 3)}^{3}}$

Step 2.13.4

Factor $8$ out of $- 8 x^{3} - 24 x^{2} + 72 x + 24$ .

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Step 2.13.4.1

Factor $8$ out of $- 8 x^{3}$ .

$f'' (x) = - \frac{8 (- x^{3}) - 24 x^{2} + 72 x + 24}{{(x^{2} + 3)}^{3}}$

Step 2.13.4.2

Factor $8$ out of $- 24 x^{2}$ .

$f'' (x) = - \frac{8 (- x^{3}) + 8 (- 3 x^{2}) + 72 x + 24}{{(x^{2} + 3)}^{3}}$

Step 2.13.4.3

Factor $8$ out of $72 x$ .

$f'' (x) = - \frac{8 (- x^{3}) + 8 (- 3 x^{2}) + 8 (9 x) + 24}{{(x^{2} + 3)}^{3}}$

Step 2.13.4.4

Factor $8$ out of $24$ .

$f'' (x) = - \frac{8 (- x^{3}) + 8 (- 3 x^{2}) + 8 (9 x) + 8 (3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.4.5

Factor $8$ out of $8 (- x^{3}) + 8 (- 3 x^{2})$ .

$f'' (x) = - \frac{8 (- x^{3} - 3 x^{2}) + 8 (9 x) + 8 (3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.4.6

Factor $8$ out of $8 (- x^{3} - 3 x^{2}) + 8 (9 x)$ .

$f'' (x) = - \frac{8 (- x^{3} - 3 x^{2} + 9 x) + 8 (3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.4.7

Factor $8$ out of $8 (- x^{3} - 3 x^{2} + 9 x) + 8 (3)$ .

$f'' (x) = - \frac{8 (- x^{3} - 3 x^{2} + 9 x + 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.5

Factor $- 1$ out of $- x^{3}$ .

$f'' (x) = - \frac{8 (- (x^{3}) - 3 x^{2} + 9 x + 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.6

Factor $- 1$ out of $- 3 x^{2}$ .

$f'' (x) = - \frac{8 (- (x^{3}) - (3 x^{2}) + 9 x + 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.7

Factor $- 1$ out of $- (x^{3}) - (3 x^{2})$ .

$f'' (x) = - \frac{8 (- (x^{3} + 3 x^{2}) + 9 x + 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.8

Factor $- 1$ out of $9 x$ .

$f'' (x) = - \frac{8 (- (x^{3} + 3 x^{2}) - (- 9 x) + 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.9

Factor $- 1$ out of $- (x^{3} + 3 x^{2}) - (- 9 x)$ .

$f'' (x) = - \frac{8 (- (x^{3} + 3 x^{2} - 9 x) + 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.10

Rewrite $3$ as $- 1 (- 3)$ .

$f'' (x) = - \frac{8 (- (x^{3} + 3 x^{2} - 9 x) - 1 \cdot - 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.11

Factor $- 1$ out of $- (x^{3} + 3 x^{2} - 9 x) - 1 (- 3)$ .

$f'' (x) = - \frac{8 (- (x^{3} + 3 x^{2} - 9 x - 3))}{{(x^{2} + 3)}^{3}}$

Step 2.13.12

Rewrite $- (x^{3} + 3 x^{2} - 9 x - 3)$ as $- 1 (x^{3} + 3 x^{2} - 9 x - 3)$ .

$f'' (x) = - \frac{8 (- 1 (x^{3} + 3 x^{2} - 9 x - 3))}{{(x^{2} + 3)}^{3}}$

Step 2.13.13

Move the negative in front of the fraction.

$f'' (x) = \frac{8 (x^{3} + 3 x^{2} - 9 x - 3)}{{(x^{2} + 3)}^{3}}$

Step 2.13.14

Multiply $- 1$ by $- 1$ .

$f'' (x) = 1 (\frac{8 (x^{3} + 3 x^{2} - 9 x - 3)}{{(x^{2} + 3)}^{3}})$

Step 2.13.15

Multiply $\frac{8 (x^{3} + 3 x^{2} - 9 x - 3)}{{(x^{2} + 3)}^{3}}$ by $1$ .

$f'' (x) = \frac{8 (x^{3} + 3 x^{2} - 9 x - 3)}{{(x^{2} + 3)}^{3}}$