Calculus Examples

$\frac{x^{2} + 64}{x}$

Step 1

Write $\frac{x^{2} + 64}{x}$ as a function.

$f (x) = \frac{x^{2} + 64}{x}$

Step 2

Find the $x$ values where the second derivative is equal to $0$ .

Tap for more steps...

Step 2.1

Find the second derivative.

Tap for more steps...

Step 2.1.1

Find the first derivative.

Tap for more steps...

Step 2.1.1.1

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 $f (x) = x^{2} + 64$ and $g (x) = x$ .

$\frac{x \frac{d}{d x} [x^{2} + 64] - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.2

Differentiate.

Tap for more steps...

Step 2.1.1.2.1

By the Sum Rule, the derivative of $x^{2} + 64$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [64]$ .

$\frac{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [64]) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.2.2

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

$\frac{x (2 x + \frac{d}{d x} [64]) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.2.3

Since $64$ is constant with respect to $x$ , the derivative of $64$ with respect to $x$ is $0$ .

$\frac{x (2 x + 0) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.2.4

Add $2 x$ and $0$ .

$\frac{x (2 x) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.3

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

$\frac{2 (x^{1} x) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.4

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

$\frac{2 (x^{1} x^{1}) - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.5

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

$\frac{2 x^{1 + 1} - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.6

Add $1$ and $1$ .

$\frac{2 x^{2} - (x^{2} + 64) \frac{d}{d x} [x]}{x^{2}}$

Step 2.1.1.7

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

$\frac{2 x^{2} - (x^{2} + 64) \cdot 1}{x^{2}}$

Step 2.1.1.8

Multiply $- 1$ by $1$ .

$\frac{2 x^{2} - (x^{2} + 64)}{x^{2}}$

Step 2.1.1.9

Simplify.

Tap for more steps...

Step 2.1.1.9.1

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - 1 \cdot 64}{x^{2}}$

Step 2.1.1.9.2

Simplify the numerator.

Tap for more steps...

Step 2.1.1.9.2.1

Multiply $- 1$ by $64$ .

$\frac{2 x^{2} - x^{2} - 64}{x^{2}}$

Step 2.1.1.9.2.2

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

$\frac{x^{2} - 64}{x^{2}}$

Step 2.1.1.9.3

Simplify the numerator.

Tap for more steps...

Step 2.1.1.9.3.1

Rewrite $64$ as $8^{2}$ .

$\frac{x^{2} - 8^{2}}{x^{2}}$

Step 2.1.1.9.3.2

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 $b = 8$ .

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

Step 2.1.2

Find the second derivative.

Tap for more steps...

Step 2.1.2.1

$\frac{x^{2} \frac{d}{d x} [(x + 8) (x - 8)] - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 2.1.2.2

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

Tap for more steps...

Step 2.1.2.2.1

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

$\frac{x^{2} \frac{d}{d x} [(x + 8) (x - 8)] - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 2.1.2.2.2

Multiply $2$ by $2$ .

$\frac{x^{2} \frac{d}{d x} [(x + 8) (x - 8)] - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.3

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 $f (x) = x + 8$ and $g (x) = x - 8$ .

$\frac{x^{2} ((x + 8) \frac{d}{d x} [x - 8] + (x - 8) \frac{d}{d x} [x + 8]) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4

Differentiate.

Tap for more steps...

Step 2.1.2.4.1

By the Sum Rule, the derivative of $x - 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 8]$ .

$\frac{x^{2} ((x + 8) (\frac{d}{d x} [x] + \frac{d}{d x} [- 8]) + (x - 8) \frac{d}{d x} [x + 8]) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.2

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

$\frac{x^{2} ((x + 8) (1 + \frac{d}{d x} [- 8]) + (x - 8) \frac{d}{d x} [x + 8]) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.3

Since $- 8$ is constant with respect to $x$ , the derivative of $- 8$ with respect to $x$ is $0$ .

$\frac{x^{2} ((x + 8) (1 + 0) + (x - 8) \frac{d}{d x} [x + 8]) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.4

Simplify the expression.

Tap for more steps...

Step 2.1.2.4.4.1

Add $1$ and $0$ .

$\frac{x^{2} ((x + 8) \cdot 1 + (x - 8) \frac{d}{d x} [x + 8]) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.4.2

Multiply $x + 8$ by $1$ .

$\frac{x^{2} (x + 8 + (x - 8) \frac{d}{d x} [x + 8]) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.5

By the Sum Rule, the derivative of $x + 8$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [8]$ .

$\frac{x^{2} (x + 8 + (x - 8) (\frac{d}{d x} [x] + \frac{d}{d x} [8])) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.6

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

$\frac{x^{2} (x + 8 + (x - 8) (1 + \frac{d}{d x} [8])) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.7

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$\frac{x^{2} (x + 8 + (x - 8) (1 + 0)) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.8

Simplify by adding terms.

Tap for more steps...

Step 2.1.2.4.8.1

Add $1$ and $0$ .

$\frac{x^{2} (x + 8 + (x - 8) \cdot 1) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.8.2

Multiply $x - 8$ by $1$ .

$\frac{x^{2} (x + 8 + x - 8) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.8.3

Add $x$ and $x$ .

$\frac{x^{2} (2 x + 8 - 8) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.8.4

Simplify by subtracting numbers.

Tap for more steps...

Step 2.1.2.4.8.4.1

Subtract $8$ from $8$ .

$\frac{x^{2} (2 x + 0) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.4.8.4.2

Add $2 x$ and $0$ .

$\frac{x^{2} (2 x) - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.5

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

Tap for more steps...

Step 2.1.2.5.1

Move $x$ .

$\frac{x \cdot x^{2} \cdot 2 - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.5.2

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

Tap for more steps...

Step 2.1.2.5.2.1

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

$\frac{x^{1} x^{2} \cdot 2 - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.5.2.2

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

$\frac{x^{1 + 2} \cdot 2 - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.5.3

Add $1$ and $2$ .

$\frac{x^{3} \cdot 2 - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.6

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

$\frac{2 \cdot x^{3} - (x + 8) (x - 8) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 2.1.2.7

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

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

Step 2.1.2.8

Multiply $2$ by $- 1$ .

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

Step 2.1.2.9

Simplify.

Tap for more steps...

Step 2.1.2.9.1

Apply the distributive property.

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

Step 2.1.2.9.2

Simplify the numerator.

Tap for more steps...

Step 2.1.2.9.2.1

Simplify each term.

Tap for more steps...

Step 2.1.2.9.2.1.1

Multiply $- 2$ by $8$ .

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

Step 2.1.2.9.2.1.2

Expand $(- 2 x - 16) (x - 8)$ using the FOIL Method.

Tap for more steps...

Step 2.1.2.9.2.1.2.1

Apply the distributive property.

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

Step 2.1.2.9.2.1.2.2

Apply the distributive property.

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

Step 2.1.2.9.2.1.2.3

Apply the distributive property.

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

Step 2.1.2.9.2.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.1.2.9.2.1.3.1

Simplify each term.

Tap for more steps...

Step 2.1.2.9.2.1.3.1.1

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

Tap for more steps...

Step 2.1.2.9.2.1.3.1.1.1

Move $x$ .

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

Step 2.1.2.9.2.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 2.1.2.9.2.1.3.1.2

Multiply $- 8$ by $- 2$ .

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

Step 2.1.2.9.2.1.3.1.3

Multiply $- 16$ by $- 8$ .

$\frac{2 x^{3} + (- 2 x^{2} + 16 x - 16 x + 128) x}{x^{4}}$

Step 2.1.2.9.2.1.3.2

Subtract $16 x$ from $16 x$ .

$\frac{2 x^{3} + (- 2 x^{2} + 0 + 128) x}{x^{4}}$

Step 2.1.2.9.2.1.3.3

Add $- 2 x^{2}$ and $0$ .

$\frac{2 x^{3} + (- 2 x^{2} + 128) x}{x^{4}}$

Step 2.1.2.9.2.1.4

Apply the distributive property.

$\frac{2 x^{3} - 2 x^{2} x + 128 x}{x^{4}}$

Step 2.1.2.9.2.1.5

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

Tap for more steps...

Step 2.1.2.9.2.1.5.1

Move $x$ .

$\frac{2 x^{3} - 2 (x \cdot x^{2}) + 128 x}{x^{4}}$

Step 2.1.2.9.2.1.5.2

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

Tap for more steps...

Step 2.1.2.9.2.1.5.2.1

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

$\frac{2 x^{3} - 2 (x^{1} x^{2}) + 128 x}{x^{4}}$

Step 2.1.2.9.2.1.5.2.2

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

$\frac{2 x^{3} - 2 x^{1 + 2} + 128 x}{x^{4}}$

Step 2.1.2.9.2.1.5.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 2 x^{3} + 128 x}{x^{4}}$

Step 2.1.2.9.2.2

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{0 + 128 x}{x^{4}}$

Step 2.1.2.9.2.3

Add $0$ and $128 x$ .

$\frac{128 x}{x^{4}}$

Step 2.1.2.9.3

Cancel the common factor of $x$ and $x^{4}$ .

Tap for more steps...

Step 2.1.2.9.3.1

Factor $x$ out of $128 x$ .

$\frac{x \cdot 128}{x^{4}}$

Step 2.1.2.9.3.2

Cancel the common factors.

Tap for more steps...

Step 2.1.2.9.3.2.1

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

$\frac{x \cdot 128}{x \cdot x^{3}}$

Step 2.1.2.9.3.2.2

Cancel the common factor.

$\frac{x \cdot 128}{x \cdot x^{3}}$

Step 2.1.2.9.3.2.3

Rewrite the expression.

$f'' (x) = \frac{128}{x^{3}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $\frac{128}{x^{3}}$ .

$\frac{128}{x^{3}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $\frac{128}{x^{3}} = 0$ .

Tap for more steps...

Step 2.2.1

Set the second derivative equal to $0$ .

$\frac{128}{x^{3}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$128 = 0$

Step 2.2.3

Since $128 \neq 0$ , there are no solutions.

No solution

Step 3

Find the domain of $f (x) = \frac{x^{2} + 64}{x}$ .

Tap for more steps...

Step 3.1

Set the denominator in $\frac{x^{2} + 64}{x}$ equal to $0$ to find where the expression is undefined.

$x = 0$

Step 3.2

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4

Create intervals around the $x$ -values where the second derivative is zero or undefined.

$(- \infty, 0) \cup (0, \infty)$

Step 5

Substitute any number from the interval $(- \infty, 0)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 5.1

Replace the variable $x$ with $- 2$ in the expression.

$f'' (- 2) = \frac{128}{{(- 2)}^{3}}$

Step 5.2

Simplify the result.

Tap for more steps...

Step 5.2.1

Raise $- 2$ to the power of $3$ .

$f'' (- 2) = \frac{128}{- 8}$

Step 5.2.2

Divide $128$ by $- 8$ .

$f'' (- 2) = - 16$

Step 5.2.3

The final answer is $- 16$ .

$- 16$

Step 5.3

The graph is concave down on the interval $(- \infty, 0)$ because $f'' (- 2)$ is negative.

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Step 6

Substitute any number from the interval $(0, \infty)$ into the second derivative and evaluate to determine the concavity.

Tap for more steps...

Step 6.1

Replace the variable $x$ with $2$ in the expression.

$f'' (2) = \frac{128}{{(2)}^{3}}$

Step 6.2

Simplify the result.

Tap for more steps...

Step 6.2.1

Raise $2$ to the power of $3$ .

$f'' (2) = \frac{128}{8}$

Step 6.2.2

Divide $128$ by $8$ .

$f'' (2) = 16$

Step 6.2.3

The final answer is $16$ .

$16$

Step 6.3

The graph is concave up on the interval $(0, \infty)$ because $f'' (2)$ is positive.

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 7

The graph is concave down when the second derivative is negative and concave up when the second derivative is positive.

Concave down on $(- \infty, 0)$ since $f'' (x)$ is negative

Concave up on $(0, \infty)$ since $f'' (x)$ is positive

Step 8