Calculus Examples

$f (x) = π - \sqrt[3]{x}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

$\frac{d}{d x} [π] + \frac{d}{d x} [- \sqrt[3]{x}]$

Step 1.1.2

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

$0 + \frac{d}{d x} [- \sqrt[3]{x}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \sqrt[3]{x}]$ .

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Step 1.2.1

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

$0 + \frac{d}{d x} [- x^{\frac{1}{3}}]$

Step 1.2.2

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

$0 - \frac{d}{d x} [x^{\frac{1}{3}}]$

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

Combine the numerators over the common denominator.

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

Step 1.2.7

Simplify the numerator.

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Step 1.2.7.1

Multiply $- 1$ by $3$ .

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

Step 1.2.7.2

Subtract $3$ from $1$ .

$0 - (\frac{1}{3} x^{\frac{- 2}{3}})$

Step 1.2.8

Move the negative in front of the fraction.

$0 - (\frac{1}{3} x^{- \frac{2}{3}})$

Step 1.2.9

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

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

Step 1.2.10

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

$0 - \frac{1}{3 x^{\frac{2}{3}}}$

Step 1.3

Subtract $\frac{1}{3 x^{\frac{2}{3}}}$ from $0$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

$- \frac{1}{3} \frac{d}{d x} [x^{\frac{2}{3} \cdot - 1}]$

Step 2.2.2.2

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 2.2.2.2.1

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

$- \frac{1}{3} \frac{d}{d x} [x^{\frac{2 \cdot - 1}{3}}]$

Step 2.2.2.2.2

Multiply $2$ by $- 1$ .

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

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

Step 2.7.2

Subtract $3$ from $- 2$ .

$- \frac{1}{3} (- \frac{2}{3} x^{\frac{- 5}{3}})$

Step 2.8

Move the negative in front of the fraction.

$- \frac{1}{3} (- \frac{2}{3} x^{- \frac{5}{3}})$

Step 2.9

Combine $x^{- \frac{5}{3}}$ and $\frac{2}{3}$ .

$- \frac{1}{3} (- \frac{x^{- \frac{5}{3}} \cdot 2}{3})$

Step 2.10

Multiply.

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Step 2.10.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{3}) \frac{x^{- \frac{5}{3}} \cdot 2}{3}$

Step 2.10.2

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3} \cdot \frac{x^{- \frac{5}{3}} \cdot 2}{3}$

Step 2.11

Multiply $\frac{1}{3}$ by $\frac{x^{- \frac{5}{3}} \cdot 2}{3}$ .

$\frac{x^{- \frac{5}{3}} \cdot 2}{3 \cdot 3}$

Step 2.12

Simplify the expression.

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Step 2.12.1

Multiply $3$ by $3$ .

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

Step 2.12.2

Move $2$ to the left of $x^{- \frac{5}{3}}$ .

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

Step 2.12.3

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

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

Step 3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2}{9 x^{\frac{5}{3}}}$ .

$\frac{2}{9 x^{\frac{5}{3}}}$