Calculus Examples

$x^{\frac{4}{3}} + 4 x^{\frac{1}{3}}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $x^{\frac{4}{3}} + 4 x^{\frac{1}{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{4}{3}}] + \frac{d}{d x} [4 x^{\frac{1}{3}}]$ .

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

Step 1.2

Evaluate $\frac{d}{d x} [x^{\frac{4}{3}}]$ .

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 1$ by $3$ .

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

Step 1.2.5.2

Subtract $3$ from $4$ .

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

Step 1.3

Evaluate $\frac{d}{d x} [4 x^{\frac{1}{3}}]$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

Combine the numerators over the common denominator.

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

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.3.6.2

Subtract $3$ from $1$ .

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

Step 1.3.7

Move the negative in front of the fraction.

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

Step 1.3.8

Combine $\frac{1}{3}$ and $x^{- \frac{2}{3}}$ .

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

Step 1.3.9

Combine $4$ and $\frac{x^{- \frac{2}{3}}}{3}$ .

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

Step 1.3.10

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 1.4

Combine $\frac{4}{3}$ and $x^{\frac{1}{3}}$ .

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{4 x^{\frac{1}{3}}}{3} + \frac{4}{3 x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{d}{d x} [\frac{4 x^{\frac{1}{3}}}{3}] + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$ .

$\frac{d}{d x} [\frac{4 x^{\frac{1}{3}}}{3}] + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{4 x^{\frac{1}{3}}}{3}]$ .

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Step 2.2.1

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

$\frac{4}{3} \frac{d}{d x} [x^{\frac{1}{3}}] + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.2.2

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

$\frac{4}{3} (\frac{1}{3} x^{\frac{1}{3} - 1}) + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.2.3

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

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

Step 2.2.4

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

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

Step 2.2.5

Combine the numerators over the common denominator.

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

Step 2.2.6

Simplify the numerator.

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Step 2.2.6.1

Multiply $- 1$ by $3$ .

$\frac{4}{3} (\frac{1}{3} x^{\frac{1 - 3}{3}}) + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.2.6.2

Subtract $3$ from $1$ .

$\frac{4}{3} (\frac{1}{3} x^{\frac{- 2}{3}}) + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.2.7

Move the negative in front of the fraction.

$\frac{4}{3} (\frac{1}{3} x^{- \frac{2}{3}}) + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.2.8

Combine $\frac{1}{3}$ and $x^{- \frac{2}{3}}$ .

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

Step 2.2.9

Multiply $\frac{4}{3}$ by $\frac{x^{- \frac{2}{3}}}{3}$ .

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

Step 2.2.10

Multiply $3$ by $3$ .

$\frac{4 x^{- \frac{2}{3}}}{9} + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.2.11

Move $x^{- \frac{2}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{4}{3 x^{\frac{2}{3}}}]$ .

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Step 2.3.1

Since $\frac{4}{3}$ is constant with respect to $x$ , the derivative of $\frac{4}{3 x^{\frac{2}{3}}}$ with respect to $x$ is $\frac{4}{3} \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} \frac{d}{d x} [\frac{1}{x^{\frac{2}{3}}}]$

Step 2.3.2

Rewrite $\frac{1}{x^{\frac{2}{3}}}$ as ${(x^{\frac{2}{3}})}^{- 1}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} \frac{d}{d x} [{(x^{\frac{2}{3}})}^{- 1}]$

Step 2.3.3

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 $g (x) = x^{\frac{2}{3}}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{2}{3}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{\frac{2}{3}}])$

Step 2.3.3.2

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

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- u^{- 2} \frac{d}{d x} [x^{\frac{2}{3}}])$

Step 2.3.3.3

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- {(x^{\frac{2}{3}})}^{- 2} \frac{d}{d x} [x^{\frac{2}{3}}])$

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- {(x^{\frac{2}{3}})}^{- 2} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.3.5

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

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Step 2.3.5.1

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

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{\frac{2}{3} \cdot - 2} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.3.5.2

Multiply $\frac{2}{3} \cdot - 2$ .

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Step 2.3.5.2.1

Combine $\frac{2}{3}$ and $- 2$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{\frac{2 \cdot - 2}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.3.5.2.2

Multiply $2$ by $- 2$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{\frac{- 4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.3.5.3

Move the negative in front of the fraction.

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1}))$

Step 2.3.6

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

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}}))$

Step 2.3.7

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

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 2.3.8

Combine the numerators over the common denominator.

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}}))$

Step 2.3.9

Simplify the numerator.

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Step 2.3.9.1

Multiply $- 1$ by $3$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{2 - 3}{3}}))$

Step 2.3.9.2

Subtract $3$ from $2$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{\frac{- 1}{3}}))$

Step 2.3.10

Move the negative in front of the fraction.

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} (\frac{2}{3} x^{- \frac{1}{3}}))$

Step 2.3.11

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- x^{- \frac{4}{3}} \frac{2 x^{- \frac{1}{3}}}{3})$

Step 2.3.12

Combine $\frac{2 x^{- \frac{1}{3}}}{3}$ and $x^{- \frac{4}{3}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- \frac{2 x^{- \frac{1}{3}} x^{- \frac{4}{3}}}{3})$

Step 2.3.13

Multiply $x^{- \frac{1}{3}}$ by $x^{- \frac{4}{3}}$ by adding the exponents.

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Step 2.3.13.1

Move $x^{- \frac{4}{3}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- \frac{2 (x^{- \frac{4}{3}} x^{- \frac{1}{3}})}{3})$

Step 2.3.13.2

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

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- \frac{2 x^{- \frac{4}{3} - \frac{1}{3}}}{3})$

Step 2.3.13.3

Combine the numerators over the common denominator.

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- \frac{2 x^{\frac{- 4 - 1}{3}}}{3})$

Step 2.3.13.4

Subtract $1$ from $- 4$ .

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- \frac{2 x^{\frac{- 5}{3}}}{3})$

Step 2.3.13.5

Move the negative in front of the fraction.

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- \frac{2 x^{- \frac{5}{3}}}{3})$

Step 2.3.14

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

$\frac{4}{9 x^{\frac{2}{3}}} + \frac{4}{3} (- \frac{2}{3 x^{\frac{5}{3}}})$

Step 2.3.15

Multiply $\frac{4}{3}$ by $\frac{2}{3 x^{\frac{5}{3}}}$ .

$\frac{4}{9 x^{\frac{2}{3}}} - \frac{4 \cdot 2}{3 (3 x^{\frac{5}{3}})}$

Step 2.3.16

Multiply $4$ by $2$ .

$\frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{3 (3 x^{\frac{5}{3}})}$

Step 2.3.17

Multiply $3$ by $3$ .

$f'' (x) = \frac{4}{9 x^{\frac{2}{3}}} - \frac{8}{9 x^{\frac{5}{3}}}$