Calculus Examples

$y = (\frac{4}{x} + x) (\frac{4}{x} - x)$

Step 1

Write $y = (\frac{4}{x} + x) (\frac{4}{x} - x)$ as a function.

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

Step 2

Find the first derivative of the function.

Tap for more steps...

Step 2.1

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 $f (x) = \frac{4}{x} + x$ and $g (x) = \frac{4}{x} - x$ .

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

Step 2.2

Differentiate.

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x}]$ .

$(\frac{4}{x} + x) (4 \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [- x]) + (\frac{4}{x} - x) \frac{d}{d x} [\frac{4}{x} + x]$

Step 2.2.3

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.2.4

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

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

Step 2.2.5

Multiply $- 1$ by $4$ .

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

Multiply $- 1$ by $1$ .

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

Step 2.2.9

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

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

Step 2.2.10

Since $4$ is constant with respect to $x$ , the derivative of $\frac{4}{x}$ with respect to $x$ is $4 \frac{d}{d x} [\frac{1}{x}]$ .

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

Step 2.2.11

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 2.2.12

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

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

Step 2.2.13

Multiply $- 1$ by $4$ .

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

Step 2.2.14

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

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

Step 2.3

Simplify.

Tap for more steps...

Step 2.3.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.3.3

Combine terms.

Tap for more steps...

Step 2.3.3.1

Combine $- 4$ and $\frac{1}{x^{2}}$ .

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

Step 2.3.3.2

Move the negative in front of the fraction.

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

Step 2.3.3.3

Combine $- 4$ and $\frac{1}{x^{2}}$ .

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

Step 2.3.3.4

Move the negative in front of the fraction.

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

Step 2.3.4

Reorder terms.

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

Step 2.3.5

Simplify each term.

Tap for more steps...

Step 2.3.5.1

Expand $(- \frac{4}{x^{2}} - 1) (\frac{4}{x} + x)$ using the FOIL Method.

Tap for more steps...

Step 2.3.5.1.1

Apply the distributive property.

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

Step 2.3.5.1.2

Apply the distributive property.

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

Step 2.3.5.1.3

Apply the distributive property.

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

Step 2.3.5.2

Simplify and combine like terms.

Tap for more steps...

Step 2.3.5.2.1

Simplify each term.

Tap for more steps...

Step 2.3.5.2.1.1

Multiply $- \frac{4}{x^{2}} \cdot \frac{4}{x}$ .

Tap for more steps...

Step 2.3.5.2.1.1.1

Multiply $\frac{4}{x}$ by $\frac{4}{x^{2}}$ .

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

Step 2.3.5.2.1.1.2

Multiply $4$ by $4$ .

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

Step 2.3.5.2.1.1.3

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

Tap for more steps...

Step 2.3.5.2.1.1.3.1

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

Tap for more steps...

Step 2.3.5.2.1.1.3.1.1

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

$- \frac{16}{x^{1} x^{2}} - \frac{4}{x^{2}} x - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.1.3.1.2

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

$- \frac{16}{x^{1 + 2}} - \frac{4}{x^{2}} x - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.1.3.2

Add $1$ and $2$ .

$- \frac{16}{x^{3}} - \frac{4}{x^{2}} x - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.5.2.1.2.1

Move the leading negative in $- \frac{4}{x^{2}}$ into the numerator.

$- \frac{16}{x^{3}} + \frac{- 4}{x^{2}} x - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.2.2

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

$- \frac{16}{x^{3}} + \frac{- 4}{x \cdot x} x - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.2.3

Cancel the common factor.

$- \frac{16}{x^{3}} + \frac{- 4}{x \cdot x} x - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.2.4

Rewrite the expression.

$- \frac{16}{x^{3}} + \frac{- 4}{x} - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.3

Move the negative in front of the fraction.

$- \frac{16}{x^{3}} - \frac{4}{x} - 1 \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.4

Rewrite $- 1 \frac{4}{x}$ as $- \frac{4}{x}$ .

$- \frac{16}{x^{3}} - \frac{4}{x} - \frac{4}{x} - 1 x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.1.5

Rewrite $- 1 x$ as $- x$ .

$- \frac{16}{x^{3}} - \frac{4}{x} - \frac{4}{x} - x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.2.2

Subtract $\frac{4}{x}$ from $- \frac{4}{x}$ .

$- \frac{16}{x^{3}} - 2 \frac{4}{x} - x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.3

Simplify each term.

Tap for more steps...

Step 2.3.5.3.1

Multiply $- 2 \frac{4}{x}$ .

Tap for more steps...

Step 2.3.5.3.1.1

Combine $- 2$ and $\frac{4}{x}$ .

$- \frac{16}{x^{3}} + \frac{- 2 \cdot 4}{x} - x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.3.1.2

Multiply $- 2$ by $4$ .

$- \frac{16}{x^{3}} + \frac{- 8}{x} - x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.3.2

Move the negative in front of the fraction.

$- \frac{16}{x^{3}} - \frac{8}{x} - x + (- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$

Step 2.3.5.4

Expand $(- \frac{4}{x^{2}} + 1) (\frac{4}{x} - x)$ using the FOIL Method.

Tap for more steps...

Step 2.3.5.4.1

Apply the distributive property.

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{4}{x^{2}} (\frac{4}{x} - x) + 1 (\frac{4}{x} - x)$

Step 2.3.5.4.2

Apply the distributive property.

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{4}{x^{2}} \cdot \frac{4}{x} - \frac{4}{x^{2}} (- x) + 1 (\frac{4}{x} - x)$

Step 2.3.5.4.3

Apply the distributive property.

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{4}{x^{2}} \cdot \frac{4}{x} - \frac{4}{x^{2}} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5

Simplify and combine like terms.

Tap for more steps...

Step 2.3.5.5.1

Simplify each term.

Tap for more steps...

Step 2.3.5.5.1.1

Multiply $- \frac{4}{x^{2}} \cdot \frac{4}{x}$ .

Tap for more steps...

Step 2.3.5.5.1.1.1

Multiply $\frac{4}{x}$ by $\frac{4}{x^{2}}$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{4 \cdot 4}{x \cdot x^{2}} - \frac{4}{x^{2}} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.1.2

Multiply $4$ by $4$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x \cdot x^{2}} - \frac{4}{x^{2}} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.1.3

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

Tap for more steps...

Step 2.3.5.5.1.1.3.1

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

Tap for more steps...

Step 2.3.5.5.1.1.3.1.1

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

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{1} x^{2}} - \frac{4}{x^{2}} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.1.3.1.2

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

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{1 + 2}} - \frac{4}{x^{2}} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.1.3.2

Add $1$ and $2$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} - \frac{4}{x^{2}} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.2

Cancel the common factor of $x$ .

Tap for more steps...

Step 2.3.5.5.1.2.1

Move the leading negative in $- \frac{4}{x^{2}}$ into the numerator.

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{- 4}{x^{2}} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.2.2

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

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{- 4}{x \cdot x} (- x) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.2.3

Factor $x$ out of $- x$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{- 4}{x \cdot x} (x \cdot - 1) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.2.4

Cancel the common factor.

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{- 4}{x \cdot x} (x \cdot - 1) + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.2.5

Rewrite the expression.

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{- 4}{x} \cdot - 1 + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.3

Combine $\frac{- 4}{x}$ and $- 1$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{- 4 \cdot - 1}{x} + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.4

Multiply $- 4$ by $- 1$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{4}{x} + 1 \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.5

Multiply $\frac{4}{x}$ by $1$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{4}{x} + \frac{4}{x} + 1 (- x)$

Step 2.3.5.5.1.6

Multiply $- x$ by $1$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{4}{x} + \frac{4}{x} - x$

Step 2.3.5.5.2

Add $\frac{4}{x}$ and $\frac{4}{x}$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + 2 \frac{4}{x} - x$

Step 2.3.5.6

Multiply $2 \frac{4}{x}$ .

Tap for more steps...

Step 2.3.5.6.1

Combine $2$ and $\frac{4}{x}$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{2 \cdot 4}{x} - x$

Step 2.3.5.6.2

Multiply $2$ by $4$ .

$- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{8}{x} - x$

Step 2.3.6

Combine the opposite terms in $- \frac{16}{x^{3}} - \frac{8}{x} - x - \frac{16}{x^{3}} + \frac{8}{x} - x$ .

Tap for more steps...

Step 2.3.6.1

Add $- \frac{8}{x}$ and $\frac{8}{x}$ .

$- \frac{16}{x^{3}} - x - \frac{16}{x^{3}} + 0 - x$

Step 2.3.6.2

Add $- \frac{16}{x^{3}} - x - \frac{16}{x^{3}}$ and $0$ .

$- \frac{16}{x^{3}} - x - \frac{16}{x^{3}} - x$

Step 2.3.7

Combine the numerators over the common denominator.

$- x - x + \frac{- 16 - 16}{x^{3}}$

Step 2.3.8

Subtract $16$ from $- 16$ .

$- x - x + \frac{- 32}{x^{3}}$

Step 2.3.9

Move the negative in front of the fraction.

$- x - x - \frac{32}{x^{3}}$

Step 2.3.10

Subtract $x$ from $- x$ .

$- 2 x - \frac{32}{x^{3}}$

Step 3

Find the second derivative of the function.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $- 2 x - \frac{32}{x^{3}}$ with respect to $x$ is $\frac{d}{d x} [- 2 x] + \frac{d}{d x} [- \frac{32}{x^{3}}]$ .

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

Step 3.2

Evaluate $\frac{d}{d x} [- 2 x]$ .

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $- 2$ by $1$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{32}{x^{3}}]$ .

Tap for more steps...

Step 3.3.1

Since $- 32$ is constant with respect to $x$ , the derivative of $- \frac{32}{x^{3}}$ with respect to $x$ is $- 32 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$f'' (x) = - 2 - 32 \frac{d}{d x} \frac{1}{x^{3}}$

Step 3.3.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$f'' (x) = - 2 - 32 \frac{d}{d x} {(x^{3})}^{- 1}$

Step 3.3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{3}$ .

Tap for more steps...

Step 3.3.3.1

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

$f'' (x) = - 2 - 32 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{3}))$

Step 3.3.3.2

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

$f'' (x) = - 2 - 32 (- u^{- 2} \frac{d}{d x} x^{3})$

Step 3.3.3.3

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

$f'' (x) = - 2 - 32 (- {(x^{3})}^{- 2} \frac{d}{d x} x^{3})$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$f'' (x) = - 2 - 32 (- {(x^{3})}^{- 2} (3 x^{2}))$

Step 3.3.5

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

Tap for more steps...

Step 3.3.5.1

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

$f'' (x) = - 2 - 32 (- x^{3 \cdot - 2} (3 x^{2}))$

Step 3.3.5.2

Multiply $3$ by $- 2$ .

$f'' (x) = - 2 - 32 (- x^{- 6} (3 x^{2}))$

Step 3.3.6

Multiply $3$ by $- 1$ .

$f'' (x) = - 2 - 32 (- 3 x^{- 6} x^{2})$

Step 3.3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 3.3.7.1

Move $x^{2}$ .

$f'' (x) = - 2 - 32 (- 3 (x^{2} x^{- 6}))$

Step 3.3.7.2

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

$f'' (x) = - 2 - 32 (- 3 x^{2 - 6})$

Step 3.3.7.3

Subtract $6$ from $2$ .

$f'' (x) = - 2 - 32 (- 3 x^{- 4})$

Step 3.3.8

Multiply $- 3$ by $- 32$ .

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

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f'' (x) = - 2 + 96 (\frac{1}{x^{4}})$

Step 3.4.2

Combine $96$ and $\frac{1}{x^{4}}$ .

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

Step 3.4.3

Reorder terms.

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

Step 4

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

$- 2 x - \frac{32}{x^{3}} = 0$

Step 5

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

No Local Extrema

Step 6

No Local Extrema

Step 7