Calculus Examples

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

Step 1

Find the second derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

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

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

Step 1.1.2

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

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Step 1.1.2.1

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

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

Step 1.1.2.2

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

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

Step 1.1.2.3

Multiply $2$ by $1$ .

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

Step 1.1.3

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

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Step 1.1.3.1

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

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

Step 1.1.3.2

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

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

Step 1.1.3.3

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

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

Step 1.1.3.4

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

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

Step 1.1.3.5

Combine the numerators over the common denominator.

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

Step 1.1.3.6

Simplify the numerator.

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Step 1.1.3.6.1

Multiply $- 1$ by $3$ .

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

Step 1.1.3.6.2

Subtract $3$ from $2$ .

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

Step 1.1.3.7

Move the negative in front of the fraction.

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

Step 1.1.3.8

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

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

Step 1.1.3.9

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

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

Step 1.1.3.10

Multiply $2$ by $3$ .

$f' (x) = 2 + \frac{6 x^{- \frac{1}{3}}}{3}$

Step 1.1.3.11

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

$f' (x) = 2 + \frac{6}{3 x^{\frac{1}{3}}}$

Step 1.1.3.12

Factor $3$ out of $6$ .

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

Step 1.1.3.13

Cancel the common factors.

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Step 1.1.3.13.1

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

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

Step 1.1.3.13.2

Cancel the common factor.

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

Step 1.1.3.13.3

Rewrite the expression.

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

Step 1.2

Find the second derivative.

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Step 1.2.1

Differentiate.

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Step 1.2.1.1

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

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

Step 1.2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$f'' (x) = 0 + \frac{d}{d x} (\frac{2}{x^{\frac{1}{3}}})$

Step 1.2.2

Evaluate $\frac{d}{d x} [\frac{2}{x^{\frac{1}{3}}}]$ .

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Step 1.2.2.1

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

$f'' (x) = 0 + 2 \frac{d}{d x} (\frac{1}{x^{\frac{1}{3}}})$

Step 1.2.2.2

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

$f'' (x) = 0 + 2 \frac{d}{d x} ({(x^{\frac{1}{3}})}^{- 1})$

Step 1.2.2.3

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 $g (x) = x^{\frac{1}{3}}$ .

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Step 1.2.2.3.1

To apply the Chain Rule, set $u$ as $x^{\frac{1}{3}}$ .

$f'' (x) = 0 + 2 (\frac{d}{d u} (u^{- 1}) \frac{d}{d x} (x^{\frac{1}{3}}))$

Step 1.2.2.3.2

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

$f'' (x) = 0 + 2 (- u^{- 2} \frac{d}{d x} x^{\frac{1}{3}})$

Step 1.2.2.3.3

Replace all occurrences of $u$ with $x^{\frac{1}{3}}$ .

$f'' (x) = 0 + 2 (- {(x^{\frac{1}{3}})}^{- 2} \frac{d}{d x} x^{\frac{1}{3}})$

Step 1.2.2.4

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

$f'' (x) = 0 + 2 (- {(x^{\frac{1}{3}})}^{- 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 1.2.2.5

Multiply the exponents in ${(x^{\frac{1}{3}})}^{- 2}$ .

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Step 1.2.2.5.1

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

$f'' (x) = 0 + 2 (- x^{\frac{1}{3} \cdot - 2} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 1.2.2.5.2

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

$f'' (x) = 0 + 2 (- x^{\frac{- 2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 1.2.2.5.3

Move the negative in front of the fraction.

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1}))$

Step 1.2.2.6

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

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} - 1 \cdot \frac{3}{3}}))$

Step 1.2.2.7

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

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}))$

Step 1.2.2.8

Combine the numerators over the common denominator.

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 1 \cdot 3}{3}}))$

Step 1.2.2.9

Simplify the numerator.

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Step 1.2.2.9.1

Multiply $- 1$ by $3$ .

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{1 - 3}{3}}))$

Step 1.2.2.9.2

Subtract $3$ from $1$ .

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{\frac{- 2}{3}}))$

Step 1.2.2.10

Move the negative in front of the fraction.

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} (\frac{1}{3} x^{- \frac{2}{3}}))$

Step 1.2.2.11

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

$f'' (x) = 0 + 2 (- x^{- \frac{2}{3}} \frac{x^{- \frac{2}{3}}}{3})$

Step 1.2.2.12

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

$f'' (x) = 0 + 2 (- \frac{x^{- \frac{2}{3}} x^{- \frac{2}{3}}}{3})$

Step 1.2.2.13

Multiply $x^{- \frac{2}{3}}$ by $x^{- \frac{2}{3}}$ by adding the exponents.

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Step 1.2.2.13.1

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

$f'' (x) = 0 + 2 (- \frac{x^{- \frac{2}{3} - \frac{2}{3}}}{3})$

Step 1.2.2.13.2

Combine the numerators over the common denominator.

$f'' (x) = 0 + 2 (- \frac{x^{\frac{- 2 - 2}{3}}}{3})$

Step 1.2.2.13.3

Subtract $2$ from $- 2$ .

$f'' (x) = 0 + 2 (- \frac{x^{\frac{- 4}{3}}}{3})$

Step 1.2.2.13.4

Move the negative in front of the fraction.

$f'' (x) = 0 + 2 (- \frac{x^{- \frac{4}{3}}}{3})$

Step 1.2.2.14

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

$f'' (x) = 0 + 2 (- \frac{1}{3 x^{\frac{4}{3}}})$

Step 1.2.2.15

Multiply $- 1$ by $2$ .

$f'' (x) = 0 - 2 \frac{1}{3 x^{\frac{4}{3}}}$

Step 1.2.2.16

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

$f'' (x) = 0 + \frac{- 2}{3 x^{\frac{4}{3}}}$

Step 1.2.2.17

Move the negative in front of the fraction.

$f'' (x) = 0 - \frac{2}{3 x^{\frac{4}{3}}}$

Step 1.2.3

Subtract $\frac{2}{3 x^{\frac{4}{3}}}$ from $0$ .

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

Step 1.3

The second derivative of $f (x)$ with respect to $x$ is $- \frac{2}{3 x^{\frac{4}{3}}}$ .

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

Step 2

Set the second derivative equal to $0$ then solve the equation $- \frac{2}{3 x^{\frac{4}{3}}} = 0$ .

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Step 2.1

Set the second derivative equal to $0$ .

$- \frac{2}{3 x^{\frac{4}{3}}} = 0$

Step 2.2

Set the numerator equal to zero.

$2 = 0$

Step 2.3

Since $2 \neq 0$ , there are no solutions.

No solution

Step 3

No values found that can make the second derivative equal to $0$ .

No Inflection Points