Calculus Examples

$\frac{9}{\sqrt[3]{x + 1}}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Constant Multiple Rule.

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Step 1.1.1

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

$\frac{d}{d x} [\frac{9}{{(x + 1)}^{\frac{1}{3}}}]$

Step 1.1.2

Since $9$ is constant with respect to $x$ , the derivative of $\frac{9}{{(x + 1)}^{\frac{1}{3}}}$ with respect to $x$ is $9 \frac{d}{d x} [\frac{1}{{(x + 1)}^{\frac{1}{3}}}]$ .

$9 \frac{d}{d x} [\frac{1}{{(x + 1)}^{\frac{1}{3}}}]$

Step 1.1.3

Apply basic rules of exponents.

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Step 1.1.3.1

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

$9 \frac{d}{d x} [{({(x + 1)}^{\frac{1}{3}})}^{- 1}]$

Step 1.1.3.2

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

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Step 1.1.3.2.1

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

$9 \frac{d}{d x} [{(x + 1)}^{\frac{1}{3} \cdot - 1}]$

Step 1.1.3.2.2

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

$9 \frac{d}{d x} [{(x + 1)}^{\frac{- 1}{3}}]$

Step 1.1.3.2.3

Move the negative in front of the fraction.

$9 \frac{d}{d x} [{(x + 1)}^{- \frac{1}{3}}]$

Step 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 $g (x) = x + 1$ .

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Step 1.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

$9 (\frac{d}{d u} [u^{- \frac{1}{3}}] \frac{d}{d x} [x + 1])$

Step 1.2.2

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

$9 (- \frac{1}{3} u^{- \frac{1}{3} - 1} \frac{d}{d x} [x + 1])$

Step 1.2.3

Replace all occurrences of $u$ with $x + 1$ .

$9 (- \frac{1}{3} {(x + 1)}^{- \frac{1}{3} - 1} \frac{d}{d x} [x + 1])$

Step 1.3

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

$9 (- \frac{1}{3} {(x + 1)}^{- \frac{1}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [x + 1])$

Step 1.4

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

$9 (- \frac{1}{3} {(x + 1)}^{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [x + 1])$

Step 1.5

Combine the numerators over the common denominator.

$9 (- \frac{1}{3} {(x + 1)}^{\frac{- 1 - 1 \cdot 3}{3}} \frac{d}{d x} [x + 1])$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $- 1$ by $3$ .

$9 (- \frac{1}{3} {(x + 1)}^{\frac{- 1 - 3}{3}} \frac{d}{d x} [x + 1])$

Step 1.6.2

Subtract $3$ from $- 1$ .

$9 (- \frac{1}{3} {(x + 1)}^{\frac{- 4}{3}} \frac{d}{d x} [x + 1])$

Step 1.7

Move the negative in front of the fraction.

$9 (- \frac{1}{3} {(x + 1)}^{- \frac{4}{3}} \frac{d}{d x} [x + 1])$

Step 1.8

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

$9 (- \frac{{(x + 1)}^{- \frac{4}{3}}}{3} \frac{d}{d x} [x + 1])$

Step 1.9

Simplify the expression.

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Step 1.9.1

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

$9 (- \frac{1}{3 {(x + 1)}^{\frac{4}{3}}} \frac{d}{d x} [x + 1])$

Step 1.9.2

Multiply $- 1$ by $9$ .

$- 9 (\frac{1}{3 {(x + 1)}^{\frac{4}{3}}} \frac{d}{d x} [x + 1])$

Step 1.10

Combine $\frac{1}{3 {(x + 1)}^{\frac{4}{3}}}$ and $- 9$ .

$\frac{- 9}{3 {(x + 1)}^{\frac{4}{3}}} \frac{d}{d x} [x + 1]$

Step 1.11

Factor $3$ out of $- 9$ .

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

Step 1.12

Cancel the common factors.

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Step 1.12.1

Factor $3$ out of $3 {(x + 1)}^{\frac{4}{3}}$ .

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

Step 1.12.2

Cancel the common factor.

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

Step 1.12.3

Rewrite the expression.

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

Step 1.13

Move the negative in front of the fraction.

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

Step 1.14

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

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

Step 1.15

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

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

Step 1.16

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

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

Step 1.17

Simplify the expression.

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Step 1.17.1

Add $1$ and $0$ .

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

Step 1.17.2

Multiply $- 1$ by $1$ .

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate using the Constant Multiple Rule.

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Step 2.1.1

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

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

Step 2.1.2

Apply basic rules of exponents.

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Step 2.1.2.1

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

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

Step 2.1.2.2

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

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

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

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Step 2.1.2.2.2.1

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

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

Step 2.1.2.2.2.2

Multiply $4$ by $- 1$ .

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

Step 2.1.2.2.3

Move the negative in front of the fraction.

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

Step 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 $g (x) = x + 1$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 2.2.2

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

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

Step 2.2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine the numerators over the common denominator.

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

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $- 1$ by $3$ .

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

Step 2.6.2

Subtract $3$ from $- 4$ .

$- 3 (- \frac{4}{3} {(x + 1)}^{\frac{- 7}{3}} \frac{d}{d x} [x + 1])$

Step 2.7

Move the negative in front of the fraction.

$- 3 (- \frac{4}{3} {(x + 1)}^{- \frac{7}{3}} \frac{d}{d x} [x + 1])$

Step 2.8

Combine ${(x + 1)}^{- \frac{7}{3}}$ and $\frac{4}{3}$ .

$- 3 (- \frac{{(x + 1)}^{- \frac{7}{3}} \cdot 4}{3} \frac{d}{d x} [x + 1])$

Step 2.9

Simplify the expression.

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Step 2.9.1

Move $4$ to the left of ${(x + 1)}^{- \frac{7}{3}}$ .

$- 3 (- \frac{4 \cdot {(x + 1)}^{- \frac{7}{3}}}{3} \frac{d}{d x} [x + 1])$

Step 2.9.2

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

$- 3 (- \frac{4}{3 {(x + 1)}^{\frac{7}{3}}} \frac{d}{d x} [x + 1])$

Step 2.9.3

Multiply $- 1$ by $- 3$ .

$3 (\frac{4}{3 {(x + 1)}^{\frac{7}{3}}} \frac{d}{d x} [x + 1])$

Step 2.10

Combine $\frac{4}{3 {(x + 1)}^{\frac{7}{3}}}$ and $3$ .

$\frac{4 \cdot 3}{3 {(x + 1)}^{\frac{7}{3}}} \frac{d}{d x} [x + 1]$

Step 2.11

Multiply $4$ by $3$ .

$\frac{12}{3 {(x + 1)}^{\frac{7}{3}}} \frac{d}{d x} [x + 1]$

Step 2.12

Factor $3$ out of $12$ .

$\frac{3 \cdot 4}{3 {(x + 1)}^{\frac{7}{3}}} \frac{d}{d x} [x + 1]$

Step 2.13

Cancel the common factors.

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Step 2.13.1

Factor $3$ out of $3 {(x + 1)}^{\frac{7}{3}}$ .

$\frac{3 \cdot 4}{3 ({(x + 1)}^{\frac{7}{3}})} \frac{d}{d x} [x + 1]$

Step 2.13.2

Cancel the common factor.

$\frac{3 \cdot 4}{3 {(x + 1)}^{\frac{7}{3}}} \frac{d}{d x} [x + 1]$

Step 2.13.3

Rewrite the expression.

$\frac{4}{{(x + 1)}^{\frac{7}{3}}} \frac{d}{d x} [x + 1]$

Step 2.14

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

$\frac{4}{{(x + 1)}^{\frac{7}{3}}} (\frac{d}{d x} [x] + \frac{d}{d x} [1])$

Step 2.15

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

$\frac{4}{{(x + 1)}^{\frac{7}{3}}} (1 + \frac{d}{d x} [1])$

Step 2.16

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

$\frac{4}{{(x + 1)}^{\frac{7}{3}}} (1 + 0)$

Step 2.17

Simplify the expression.

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Step 2.17.1

Add $1$ and $0$ .

$\frac{4}{{(x + 1)}^{\frac{7}{3}}} \cdot 1$

Step 2.17.2

Multiply $\frac{4}{{(x + 1)}^{\frac{7}{3}}}$ by $1$ .

$f'' (x) = \frac{4}{{(x + 1)}^{\frac{7}{3}}}$