Calculus Examples

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

Step 1

Find the first derivative of the function.

Tap for more steps...

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 $f (x) = x$ and $g (x) = x^{2} - 4$ .

$\frac{(x^{2} - 4) \frac{d}{d x} [x] - x \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.2

Differentiate.

Tap for more steps...

Step 1.2.1

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

$\frac{(x^{2} - 4) \cdot 1 - x \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.2.2

Multiply $x^{2} - 4$ by $1$ .

$\frac{x^{2} - 4 - x \frac{d}{d x} [x^{2} - 4]}{{(x^{2} - 4)}^{2}}$

Step 1.2.3

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

$\frac{x^{2} - 4 - x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{2}}$

Step 1.2.4

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

$\frac{x^{2} - 4 - x (2 x + \frac{d}{d x} [- 4])}{{(x^{2} - 4)}^{2}}$

Step 1.2.5

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

$\frac{x^{2} - 4 - x (2 x + 0)}{{(x^{2} - 4)}^{2}}$

Step 1.2.6

Simplify the expression.

Tap for more steps...

Step 1.2.6.1

Add $2 x$ and $0$ .

$\frac{x^{2} - 4 - x (2 x)}{{(x^{2} - 4)}^{2}}$

Step 1.2.6.2

Multiply $2$ by $- 1$ .

$\frac{x^{2} - 4 - 2 x \cdot x}{{(x^{2} - 4)}^{2}}$

Step 1.3

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

$\frac{x^{2} - 4 - 2 (x^{1} x)}{{(x^{2} - 4)}^{2}}$

Step 1.4

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

$\frac{x^{2} - 4 - 2 (x^{1} x^{1})}{{(x^{2} - 4)}^{2}}$

Step 1.5

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

$\frac{x^{2} - 4 - 2 x^{1 + 1}}{{(x^{2} - 4)}^{2}}$

Step 1.6

Add $1$ and $1$ .

$\frac{x^{2} - 4 - 2 x^{2}}{{(x^{2} - 4)}^{2}}$

Step 1.7

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

$\frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}}$

Step 2

Find the second derivative of the function.

Tap for more steps...

Step 2.1

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

Multiply the exponents in ${({(x^{2} - 4)}^{2})}^{2}$ .

Tap for more steps...

Step 2.2.1.1

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

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

Step 2.2.1.2

Multiply $2$ by $2$ .

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

Step 2.2.2

By the Sum Rule, the derivative of $- x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [- x^{2}] + \frac{d}{d x} [- 4]$ .

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

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $2$ by $- 1$ .

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

Step 2.2.6

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

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

Step 2.2.7

Add $- 2 x$ and $0$ .

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

Step 2.3

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 $g (x) = x^{2} - 4$ .

Tap for more steps...

Step 2.3.1

To apply the Chain Rule, set $u$ as $x^{2} - 4$ .

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

Step 2.3.2

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

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

Step 2.3.3

Replace all occurrences of $u$ with $x^{2} - 4$ .

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

Step 2.4

Differentiate.

Tap for more steps...

Step 2.4.1

Multiply $2$ by $- 1$ .

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

Step 2.4.2

By the Sum Rule, the derivative of $x^{2} - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4]$ .

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

Step 2.4.3

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

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

Step 2.4.4

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

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

Step 2.4.5

Simplify the expression.

Tap for more steps...

Step 2.4.5.1

Add $2 x$ and $0$ .

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

Step 2.4.5.2

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

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

Step 2.4.5.3

Multiply $2$ by $- 2$ .

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

Step 2.5

Simplify.

Tap for more steps...

Step 2.5.1

Apply the distributive property.

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

Step 2.5.2

Apply the distributive property.

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

Step 2.5.3

Simplify the numerator.

Tap for more steps...

Step 2.5.3.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 2.5.3.1.2

Rewrite ${(x^{2} - 4)}^{2}$ as $(x^{2} - 4) (x^{2} - 4)$ .

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

Step 2.5.3.1.3

Expand $(x^{2} - 4) (x^{2} - 4)$ using the FOIL Method.

Tap for more steps...

Step 2.5.3.1.3.1

Apply the distributive property.

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

Step 2.5.3.1.3.2

Apply the distributive property.

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

Step 2.5.3.1.3.3

Apply the distributive property.

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

Step 2.5.3.1.4

Simplify and combine like terms.

Tap for more steps...

Step 2.5.3.1.4.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.4.1.1

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

Tap for more steps...

Step 2.5.3.1.4.1.1.1

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

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

Step 2.5.3.1.4.1.1.2

Add $2$ and $2$ .

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

Step 2.5.3.1.4.1.2

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

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

Step 2.5.3.1.4.1.3

Multiply $- 4$ by $- 4$ .

$f'' (x) = \frac{- 2 (x^{4} - 4 x^{2} - 4 x^{2} + 16) x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.4.2

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

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

Step 2.5.3.1.5

Apply the distributive property.

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

Step 2.5.3.1.6

Simplify.

Tap for more steps...

Step 2.5.3.1.6.1

Multiply $- 8$ by $- 2$ .

$f'' (x) = \frac{(- 2 x^{4} + 16 x^{2} - 2 \cdot 16) x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.6.2

Multiply $- 2$ by $16$ .

$f'' (x) = \frac{(- 2 x^{4} + 16 x^{2} - 32) x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.7

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{4} x + 16 x^{2} x - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8

Simplify.

Tap for more steps...

Step 2.5.3.1.8.1

Multiply $x^{4}$ by $x$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.8.1.1

Move $x$ .

$f'' (x) = \frac{- 2 (x \cdot x^{4}) + 16 x^{2} x - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8.1.2

Multiply $x$ by $x^{4}$ .

Tap for more steps...

Step 2.5.3.1.8.1.2.1

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

$f'' (x) = \frac{- 2 (x \cdot x^{4}) + 16 x^{2} x - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8.1.2.2

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

$f'' (x) = \frac{- 2 x^{1 + 4} + 16 x^{2} x - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8.1.3

Add $1$ and $4$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{2} x - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8.2

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

Tap for more steps...

Step 2.5.3.1.8.2.1

Move $x$ .

$f'' (x) = \frac{- 2 x^{5} + 16 (x \cdot x^{2}) - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8.2.2

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

Tap for more steps...

Step 2.5.3.1.8.2.2.1

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

$f'' (x) = \frac{- 2 x^{5} + 16 (x \cdot x^{2}) - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8.2.2.2

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

$f'' (x) = \frac{- 2 x^{5} + 16 x^{1 + 2} - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.8.2.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + (- 4 (- x^{2}) - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.9

Simplify each term.

Tap for more steps...

Step 2.5.3.1.9.1

Multiply $- 1$ by $- 4$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + (4 x^{2} - 4 \cdot - 4) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.9.2

Multiply $- 4$ by $- 4$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + (4 x^{2} + 16) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.10

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

Tap for more steps...

Step 2.5.3.1.10.1

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

Tap for more steps...

Step 2.5.3.1.10.1.1

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

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + (4 x^{2} + 16) (x^{2} x - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.10.1.2

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

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + (4 x^{2} + 16) (x^{2 + 1} - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.10.2

Add $2$ and $1$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + (4 x^{2} + 16) (x^{3} - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.11

Expand $(4 x^{2} + 16) (x^{3} - 4 x)$ using the FOIL Method.

Tap for more steps...

Step 2.5.3.1.11.1

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{2} (x^{3} - 4 x) + 16 (x^{3} - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.11.2

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{2} x^{3} + 4 x^{2} (- 4 x) + 16 (x^{3} - 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.11.3

Apply the distributive property.

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{2} x^{3} + 4 x^{2} (- 4 x) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12

Simplify and combine like terms.

Tap for more steps...

Step 2.5.3.1.12.1

Simplify each term.

Tap for more steps...

Step 2.5.3.1.12.1.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 2.5.3.1.12.1.1.1

Move $x^{3}$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 (x^{3} x^{2}) + 4 x^{2} (- 4 x) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.1.2

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

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{3 + 2} + 4 x^{2} (- 4 x) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.1.3

Add $3$ and $2$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} + 4 x^{2} (- 4 x) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.2

Rewrite using the commutative property of multiplication.

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} + 4 \cdot (- 4 x^{2} x) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.3

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

Tap for more steps...

Step 2.5.3.1.12.1.3.1

Move $x$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} + 4 \cdot (- 4 (x \cdot x^{2})) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.3.2

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

Tap for more steps...

Step 2.5.3.1.12.1.3.2.1

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

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} + 4 \cdot (- 4 (x \cdot x^{2})) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.3.2.2

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

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} + 4 \cdot (- 4 x^{1 + 2}) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.3.3

Add $1$ and $2$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} + 4 \cdot (- 4 x^{3}) + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.4

Multiply $4$ by $- 4$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} - 16 x^{3} + 16 x^{3} + 16 (- 4 x)}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.1.5

Multiply $- 4$ by $16$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} - 16 x^{3} + 16 x^{3} - 64 x}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.2

Add $- 16 x^{3}$ and $16 x^{3}$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} + 0 - 64 x}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.1.12.3

Add $4 x^{5}$ and $0$ .

$f'' (x) = \frac{- 2 x^{5} + 16 x^{3} - 32 x + 4 x^{5} - 64 x}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.2

Add $- 2 x^{5}$ and $4 x^{5}$ .

$f'' (x) = \frac{2 x^{5} + 16 x^{3} - 32 x - 64 x}{{(x^{2} - 4)}^{4}}$

Step 2.5.3.3

Subtract $64 x$ from $- 32 x$ .

$f'' (x) = \frac{2 x^{5} + 16 x^{3} - 96 x}{{(x^{2} - 4)}^{4}}$

Step 2.5.4

Simplify the numerator.

Tap for more steps...

Step 2.5.4.1

Factor $2 x$ out of $2 x^{5} + 16 x^{3} - 96 x$ .

Tap for more steps...

Step 2.5.4.1.1

Factor $2 x$ out of $2 x^{5}$ .

$f'' (x) = \frac{2 x (x^{4}) + 16 x^{3} - 96 x}{{(x^{2} - 4)}^{4}}$

Step 2.5.4.1.2

Factor $2 x$ out of $16 x^{3}$ .

$f'' (x) = \frac{2 x (x^{4}) + 2 x (8 x^{2}) - 96 x}{{(x^{2} - 4)}^{4}}$

Step 2.5.4.1.3

Factor $2 x$ out of $- 96 x$ .

$f'' (x) = \frac{2 x (x^{4}) + 2 x (8 x^{2}) + 2 x (- 48)}{{(x^{2} - 4)}^{4}}$

Step 2.5.4.1.4

Factor $2 x$ out of $2 x (x^{4}) + 2 x (8 x^{2})$ .

$f'' (x) = \frac{2 x (x^{4} + 8 x^{2}) + 2 x (- 48)}{{(x^{2} - 4)}^{4}}$

Step 2.5.4.1.5

Factor $2 x$ out of $2 x (x^{4} + 8 x^{2}) + 2 x (- 48)$ .

$f'' (x) = \frac{2 x (x^{4} + 8 x^{2} - 48)}{{(x^{2} - 4)}^{4}}$

Step 2.5.4.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$f'' (x) = \frac{2 x ({(x^{2})}^{2} + 8 x^{2} - 48)}{{(x^{2} - 4)}^{4}}$

Step 2.5.4.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$f'' (x) = \frac{2 x (u^{2} + 8 u - 48)}{{(x^{2} - 4)}^{4}}$

Step 2.5.4.4

Factor $u^{2} + 8 u - 48$ using the AC method.

Tap for more steps...

Step 2.5.4.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 48$ and whose sum is $8$ .

$- 4, 12$

Step 2.5.4.4.2

Write the factored form using these integers.

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

Step 2.5.4.5

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

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

Step 2.5.4.6

Rewrite $4$ as $2^{2}$ .

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

Step 2.5.4.7

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 2.5.5

Simplify the denominator.

Tap for more steps...

Step 2.5.5.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.5.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

Step 2.5.5.3

Apply the product rule to $(x + 2) (x - 2)$ .

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

Step 2.5.6

Cancel the common factor of $x + 2$ and ${(x + 2)}^{4}$ .

Tap for more steps...

Step 2.5.6.1

Factor $x + 2$ out of $2 x (x + 2) (x - 2) (x^{2} + 12)$ .

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

Step 2.5.6.2

Cancel the common factors.

Tap for more steps...

Step 2.5.6.2.1

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

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

Step 2.5.6.2.2

Cancel the common factor.

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

Step 2.5.6.2.3

Rewrite the expression.

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

Step 2.5.7

Cancel the common factor of $x - 2$ and ${(x - 2)}^{4}$ .

Tap for more steps...

Step 2.5.7.1

Factor $x - 2$ out of $(2 x (x - 2)) (x^{2} + 12)$ .

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

Step 2.5.7.2

Cancel the common factors.

Tap for more steps...

Step 2.5.7.2.1

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

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

Step 2.5.7.2.2

Cancel the common factor.

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

Step 2.5.7.2.3

Rewrite the expression.

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

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

Step 3

To find the local maximum and minimum values of the function, set the derivative equal to $0$ and solve.

$\frac{- x^{2} - 4}{{(x^{2} - 4)}^{2}} = 0$

Step 4

Since there is no value of $x$ that makes the first derivative equal to $0$ , there are no local extrema.

No Local Extrema

Step 5

No Local Extrema

Step 6