Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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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 $n = \frac{2}{3}$ .

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

Combine the numerators over the common denominator.

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

Step 1.2.5

Simplify the numerator.

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Step 1.2.5.1

Multiply $- 1$ by $3$ .

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

Step 1.2.5.2

Subtract $3$ from $2$ .

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

Step 1.2.6

Move the negative in front of the fraction.

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

Step 1.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} + 0$

Step 1.4.2

Combine terms.

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Step 1.4.2.1

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

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

Step 1.4.2.2

Add $\frac{2}{3 x^{\frac{1}{3}}}$ and $0$ .

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

Apply basic rules of exponents.

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Step 2.2.1

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

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 2.2.2.3

Move the negative in front of the fraction.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Combine the numerators over the common denominator.

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

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $- 1$ by $3$ .

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

Step 2.7.2

Subtract $3$ from $- 1$ .

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

Step 2.8

Move the negative in front of the fraction.

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

Step 2.9

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

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

Step 2.10

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

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

Step 2.11

Multiply.

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Step 2.11.1

Multiply $3$ by $3$ .

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

Step 2.11.2

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

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