Calculus Examples

$- \frac{81}{\sqrt[3]{x + 5}}$

Step 1

Write $- \frac{81}{\sqrt[3]{x + 5}}$ as a function.

$f (x) = - \frac{81}{\sqrt[3]{x + 5}}$

Step 2

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

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Step 2.1

Find the second derivative.

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Step 2.1.1

Find the first derivative.

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Step 2.1.1.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 5}$ as ${(x + 5)}^{\frac{1}{3}}$ .

$\frac{d}{d x} [- \frac{81}{{(x + 5)}^{\frac{1}{3}}}]$

Step 2.1.1.1.2

Since $- 81$ is constant with respect to $x$ , the derivative of $- \frac{81}{{(x + 5)}^{\frac{1}{3}}}$ with respect to $x$ is $- 81 \frac{d}{d x} [\frac{1}{{(x + 5)}^{\frac{1}{3}}}]$ .

$- 81 \frac{d}{d x} [\frac{1}{{(x + 5)}^{\frac{1}{3}}}]$

Step 2.1.1.1.3

Apply basic rules of exponents.

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Step 2.1.1.1.3.1

Rewrite $\frac{1}{{(x + 5)}^{\frac{1}{3}}}$ as ${({(x + 5)}^{\frac{1}{3}})}^{- 1}$ .

$- 81 \frac{d}{d x} [{({(x + 5)}^{\frac{1}{3}})}^{- 1}]$

Step 2.1.1.1.3.2

Multiply the exponents in ${({(x + 5)}^{\frac{1}{3}})}^{- 1}$ .

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Step 2.1.1.1.3.2.1

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

$- 81 \frac{d}{d x} [{(x + 5)}^{\frac{1}{3} \cdot - 1}]$

Step 2.1.1.1.3.2.2

Combine $\frac{1}{3}$ and $- 1$ .

$- 81 \frac{d}{d x} [{(x + 5)}^{\frac{- 1}{3}}]$

Step 2.1.1.1.3.2.3

Move the negative in front of the fraction.

$- 81 \frac{d}{d x} [{(x + 5)}^{- \frac{1}{3}}]$

Step 2.1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{1}{3}}$ and $g (x) = x + 5$ .

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Step 2.1.1.2.1

To apply the Chain Rule, set $u$ as $x + 5$ .

$- 81 (\frac{d}{d u} [u^{- \frac{1}{3}}] \frac{d}{d x} [x + 5])$

Step 2.1.1.2.2

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

$- 81 (- \frac{1}{3} u^{- \frac{1}{3} - 1} \frac{d}{d x} [x + 5])$

Step 2.1.1.2.3

Replace all occurrences of $u$ with $x + 5$ .

$- 81 (- \frac{1}{3} {(x + 5)}^{- \frac{1}{3} - 1} \frac{d}{d x} [x + 5])$

Step 2.1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 81 (- \frac{1}{3} {(x + 5)}^{- \frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.1.4

Combine $- 1$ and $\frac{3}{3}$ .

$- 81 (- \frac{1}{3} {(x + 5)}^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.1.5

Combine the numerators over the common denominator.

$- 81 (- \frac{1}{3} {(x + 5)}^{\frac{- 1 - 1 \cdot 3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.1.6

Simplify the numerator.

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Step 2.1.1.6.1

Multiply $- 1$ by $3$ .

$- 81 (- \frac{1}{3} {(x + 5)}^{\frac{- 1 - 3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.1.6.2

Subtract $3$ from $- 1$ .

$- 81 (- \frac{1}{3} {(x + 5)}^{\frac{- 4}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.1.7

Move the negative in front of the fraction.

$- 81 (- \frac{1}{3} {(x + 5)}^{- \frac{4}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.1.8

Combine ${(x + 5)}^{- \frac{4}{3}}$ and $\frac{1}{3}$ .

$- 81 (- \frac{{(x + 5)}^{- \frac{4}{3}}}{3} \frac{d}{d x} [x + 5])$

Step 2.1.1.9

Simplify the expression.

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Step 2.1.1.9.1

Move ${(x + 5)}^{- \frac{4}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 81 (- \frac{1}{3 {(x + 5)}^{\frac{4}{3}}} \frac{d}{d x} [x + 5])$

Step 2.1.1.9.2

Multiply $- 1$ by $- 81$ .

$81 (\frac{1}{3 {(x + 5)}^{\frac{4}{3}}} \frac{d}{d x} [x + 5])$

Step 2.1.1.10

Combine $\frac{1}{3 {(x + 5)}^{\frac{4}{3}}}$ and $81$ .

$\frac{81}{3 {(x + 5)}^{\frac{4}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.1.11

Factor $3$ out of $81$ .

$\frac{3 \cdot 27}{3 {(x + 5)}^{\frac{4}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.1.12

Cancel the common factors.

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Step 2.1.1.12.1

Factor $3$ out of $3 {(x + 5)}^{\frac{4}{3}}$ .

$\frac{3 \cdot 27}{3 ({(x + 5)}^{\frac{4}{3}})} \frac{d}{d x} [x + 5]$

Step 2.1.1.12.2

Cancel the common factor.

$\frac{3 \cdot 27}{3 {(x + 5)}^{\frac{4}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.1.12.3

Rewrite the expression.

$\frac{27}{{(x + 5)}^{\frac{4}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.1.13

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$\frac{27}{{(x + 5)}^{\frac{4}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 2.1.1.14

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

$\frac{27}{{(x + 5)}^{\frac{4}{3}}} (1 + \frac{d}{d x} [5])$

Step 2.1.1.15

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$\frac{27}{{(x + 5)}^{\frac{4}{3}}} (1 + 0)$

Step 2.1.1.16

Simplify the expression.

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Step 2.1.1.16.1

Add $1$ and $0$ .

$\frac{27}{{(x + 5)}^{\frac{4}{3}}} \cdot 1$

Step 2.1.1.16.2

Multiply $\frac{27}{{(x + 5)}^{\frac{4}{3}}}$ by $1$ .

$f' (x) = \frac{27}{{(x + 5)}^{\frac{4}{3}}}$

Step 2.1.2

Find the second derivative.

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Step 2.1.2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.2.1.1

Since $27$ is constant with respect to $x$ , the derivative of $\frac{27}{{(x + 5)}^{\frac{4}{3}}}$ with respect to $x$ is $27 \frac{d}{d x} [\frac{1}{{(x + 5)}^{\frac{4}{3}}}]$ .

$27 \frac{d}{d x} [\frac{1}{{(x + 5)}^{\frac{4}{3}}}]$

Step 2.1.2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1.2.1

Rewrite $\frac{1}{{(x + 5)}^{\frac{4}{3}}}$ as ${({(x + 5)}^{\frac{4}{3}})}^{- 1}$ .

$27 \frac{d}{d x} [{({(x + 5)}^{\frac{4}{3}})}^{- 1}]$

Step 2.1.2.1.2.2

Multiply the exponents in ${({(x + 5)}^{\frac{4}{3}})}^{- 1}$ .

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Step 2.1.2.1.2.2.1

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

$27 \frac{d}{d x} [{(x + 5)}^{\frac{4}{3} \cdot - 1}]$

Step 2.1.2.1.2.2.2

Multiply $\frac{4}{3} \cdot - 1$ .

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Step 2.1.2.1.2.2.2.1

Combine $\frac{4}{3}$ and $- 1$ .

$27 \frac{d}{d x} [{(x + 5)}^{\frac{4 \cdot - 1}{3}}]$

Step 2.1.2.1.2.2.2.2

Multiply $4$ by $- 1$ .

$27 \frac{d}{d x} [{(x + 5)}^{\frac{- 4}{3}}]$

Step 2.1.2.1.2.2.3

Move the negative in front of the fraction.

$27 \frac{d}{d x} [{(x + 5)}^{- \frac{4}{3}}]$

Step 2.1.2.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{4}{3}}$ and $g (x) = x + 5$ .

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Step 2.1.2.2.1

To apply the Chain Rule, set $u$ as $x + 5$ .

$27 (\frac{d}{d u} [u^{- \frac{4}{3}}] \frac{d}{d x} [x + 5])$

Step 2.1.2.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{4}{3}$ .

$27 (- \frac{4}{3} u^{- \frac{4}{3} - 1} \frac{d}{d x} [x + 5])$

Step 2.1.2.2.3

Replace all occurrences of $u$ with $x + 5$ .

$27 (- \frac{4}{3} {(x + 5)}^{- \frac{4}{3} - 1} \frac{d}{d x} [x + 5])$

Step 2.1.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$27 (- \frac{4}{3} {(x + 5)}^{- \frac{4}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.2.4

Combine $- 1$ and $\frac{3}{3}$ .

$27 (- \frac{4}{3} {(x + 5)}^{- \frac{4}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.2.5

Combine the numerators over the common denominator.

$27 (- \frac{4}{3} {(x + 5)}^{\frac{- 4 - 1 \cdot 3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.2.6

Simplify the numerator.

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Step 2.1.2.6.1

Multiply $- 1$ by $3$ .

$27 (- \frac{4}{3} {(x + 5)}^{\frac{- 4 - 3}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.2.6.2

Subtract $3$ from $- 4$ .

$27 (- \frac{4}{3} {(x + 5)}^{\frac{- 7}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.2.7

Move the negative in front of the fraction.

$27 (- \frac{4}{3} {(x + 5)}^{- \frac{7}{3}} \frac{d}{d x} [x + 5])$

Step 2.1.2.8

Combine ${(x + 5)}^{- \frac{7}{3}}$ and $\frac{4}{3}$ .

$27 (- \frac{{(x + 5)}^{- \frac{7}{3}} \cdot 4}{3} \frac{d}{d x} [x + 5])$

Step 2.1.2.9

Simplify the expression.

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Step 2.1.2.9.1

Move $4$ to the left of ${(x + 5)}^{- \frac{7}{3}}$ .

$27 (- \frac{4 \cdot {(x + 5)}^{- \frac{7}{3}}}{3} \frac{d}{d x} [x + 5])$

Step 2.1.2.9.2

Move ${(x + 5)}^{- \frac{7}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$27 (- \frac{4}{3 {(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5])$

Step 2.1.2.9.3

Multiply $- 1$ by $27$ .

$- 27 (\frac{4}{3 {(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5])$

Step 2.1.2.10

Combine $\frac{4}{3 {(x + 5)}^{\frac{7}{3}}}$ and $- 27$ .

$\frac{4 \cdot - 27}{3 {(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.2.11

Multiply $4$ by $- 27$ .

$\frac{- 108}{3 {(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.2.12

Factor $3$ out of $- 108$ .

$\frac{3 \cdot - 36}{3 {(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.2.13

Cancel the common factors.

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Step 2.1.2.13.1

Factor $3$ out of $3 {(x + 5)}^{\frac{7}{3}}$ .

$\frac{3 \cdot - 36}{3 ({(x + 5)}^{\frac{7}{3}})} \frac{d}{d x} [x + 5]$

Step 2.1.2.13.2

Cancel the common factor.

$\frac{3 \cdot - 36}{3 {(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.2.13.3

Rewrite the expression.

$\frac{- 36}{{(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.2.14

Move the negative in front of the fraction.

$- \frac{36}{{(x + 5)}^{\frac{7}{3}}} \frac{d}{d x} [x + 5]$

Step 2.1.2.15

By the Sum Rule, the derivative of $x + 5$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [5]$ .

$- \frac{36}{{(x + 5)}^{\frac{7}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [5])$

Step 2.1.2.16

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

$- \frac{36}{{(x + 5)}^{\frac{7}{3}}} (1 + \frac{d}{d x} [5])$

Step 2.1.2.17

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$- \frac{36}{{(x + 5)}^{\frac{7}{3}}} (1 + 0)$

Step 2.1.2.18

Simplify the expression.

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Step 2.1.2.18.1

Add $1$ and $0$ .

$- \frac{36}{{(x + 5)}^{\frac{7}{3}}} \cdot 1$

Step 2.1.2.18.2

Multiply $- 1$ by $1$ .

$f'' (x) = - \frac{36}{{(x + 5)}^{\frac{7}{3}}}$

Step 2.1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{36}{{(x + 5)}^{\frac{7}{3}}}$ .

$- \frac{36}{{(x + 5)}^{\frac{7}{3}}}$

Step 2.2

Set the second derivative equal to $0$ then solve the equation $- \frac{36}{{(x + 5)}^{\frac{7}{3}}} = 0$ .

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Step 2.2.1

Set the second derivative equal to $0$ .

$- \frac{36}{{(x + 5)}^{\frac{7}{3}}} = 0$

Step 2.2.2

Set the numerator equal to zero.

$36 = 0$

Step 2.2.3

Since $36 \neq 0$ , there are no solutions.

No solution

Step 3

Find the domain of $f (x) = - \frac{81}{\sqrt[3]{x + 5}}$ .

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Step 3.1

Set the denominator in $\frac{81}{\sqrt[3]{x + 5}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[3]{x + 5} = 0$

Step 3.2

Solve for $x$ .

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Step 3.2.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{x + 5}}^{3} = 0^{3}$

Step 3.2.2

Simplify each side of the equation.

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Step 3.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x + 5}$ as ${(x + 5)}^{\frac{1}{3}}$ .

${({(x + 5)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Simplify ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.2.2.1.1

Multiply the exponents in ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

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Step 3.2.2.2.1.1.1

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

${(x + 5)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.2.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.2.2.1.1.2.1

Cancel the common factor.

${(x + 5)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.2.2.2.1.1.2.2

Rewrite the expression.

${(x + 5)}^{1} = 0^{3}$

Step 3.2.2.2.1.2

Simplify.

$x + 5 = 0^{3}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Raising $0$ to any positive power yields $0$ .

$x + 5 = 0$

Step 3.2.3

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 3.3

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

Interval Notation:

$(- \infty, - 5) \cup (- 5, \infty)$

Set-Builder Notation:

${x | x \neq - 5}$

Interval Notation:

$(- \infty, - 5) \cup (- 5, \infty)$

Set-Builder Notation:

${x | x \neq - 5}$

Step 4

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

$(- \infty, - 5) \cup (- 5, \infty)$

Step 5

Substitute any number from the interval $(- \infty, - 5)$ into the second derivative and evaluate to determine the concavity.

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Step 5.1

Replace the variable $x$ with $- 8$ in the expression.

$f'' (- 8) = - \frac{36}{{((- 8) + 5)}^{\frac{7}{3}}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Add $- 8$ and $5$ .

$f'' (- 8) = - \frac{36}{{(- 3)}^{\frac{7}{3}}}$

Step 5.2.2

The final answer is $- \frac{36}{{(- 3)}^{\frac{7}{3}}}$ .

$- \frac{36}{{(- 3)}^{\frac{7}{3}}}$

Step 5.3

The graph is concave up on the interval $(- \infty, - 5)$ because $f'' (- 8)$ is positive.

Concave up on $(- \infty, - 5)$ since $f'' (x)$ is positive

Step 6

Substitute any number from the interval $(- 5, \infty)$ into the second derivative and evaluate to determine the concavity.

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Step 6.1

Replace the variable $x$ with $0$ in the expression.

$f'' (0) = - \frac{36}{{((0) + 5)}^{\frac{7}{3}}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Add $0$ and $5$ .

$f'' (0) = - \frac{36}{5^{\frac{7}{3}}}$

Step 6.2.2

The final answer is $- \frac{36}{5^{\frac{7}{3}}}$ .

$- \frac{36}{5^{\frac{7}{3}}}$

Step 6.3

The graph is concave down on the interval $(- 5, \infty)$ because $f'' (0)$ is negative.

Concave down on $(- 5, \infty)$ since $f'' (x)$ is negative

Step 7

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

Concave up on $(- \infty, - 5)$ since $f'' (x)$ is positive

Concave down on $(- 5, \infty)$ since $f'' (x)$ is negative

Step 8