Calculus Examples

$y = \frac{1}{2} + \frac{1}{3} x^{5} - \frac{1}{3} x^{2} + \frac{5}{9} x$

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 $\frac{1}{2} + \frac{1}{3} x^{5} - \frac{1}{3} x^{2} + \frac{5}{9} x$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2}] + \frac{d}{d x} [\frac{1}{3} x^{5}] + \frac{d}{d x} [- \frac{1}{3} x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$ .

$\frac{d}{d x} [\frac{1}{2}] + \frac{d}{d x} [\frac{1}{3} x^{5}] + \frac{d}{d x} [- \frac{1}{3} x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{1}{2}$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [\frac{1}{3} x^{5}] + \frac{d}{d x} [- \frac{1}{3} x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{5}]$ .

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Step 1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{5}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{5}]$ .

$0 + \frac{1}{3} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- \frac{1}{3} x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$0 + \frac{1}{3} (5 x^{4}) + \frac{d}{d x} [- \frac{1}{3} x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.2.3

Combine $5$ and $\frac{1}{3}$ .

$0 + \frac{5}{3} x^{4} + \frac{d}{d x} [- \frac{1}{3} x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.2.4

Combine $\frac{5}{3}$ and $x^{4}$ .

$0 + \frac{5 x^{4}}{3} + \frac{d}{d x} [- \frac{1}{3} x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.3

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

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Step 1.3.1

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

$0 + \frac{5 x^{4}}{3} - \frac{1}{3} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.3.2

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

$0 + \frac{5 x^{4}}{3} - \frac{1}{3} (2 x) + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.3.3

Multiply $2$ by $- 1$ .

$0 + \frac{5 x^{4}}{3} - 2 (\frac{1}{3}) x + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.3.4

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

$0 + \frac{5 x^{4}}{3} + \frac{- 2}{3} x + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.3.5

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

$0 + \frac{5 x^{4}}{3} + \frac{- 2 x}{3} + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.3.6

Move the negative in front of the fraction.

$0 + \frac{5 x^{4}}{3} - \frac{2 x}{3} + \frac{d}{d x} [\frac{5}{9} x]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{5}{9} x]$ .

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Step 1.4.1

Since $\frac{5}{9}$ is constant with respect to $x$ , the derivative of $\frac{5}{9} x$ with respect to $x$ is $\frac{5}{9} \frac{d}{d x} [x]$ .

$0 + \frac{5 x^{4}}{3} - \frac{2 x}{3} + \frac{5}{9} \frac{d}{d x} [x]$

Step 1.4.2

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

$0 + \frac{5 x^{4}}{3} - \frac{2 x}{3} + \frac{5}{9} \cdot 1$

Step 1.4.3

Multiply $\frac{5}{9}$ by $1$ .

$0 + \frac{5 x^{4}}{3} - \frac{2 x}{3} + \frac{5}{9}$

Step 1.5

Add $0$ and $\frac{5 x^{4}}{3}$ .

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{5 x^{4}}{3} - \frac{2 x}{3} + \frac{5}{9}$ with respect to $x$ is $\frac{d}{d x} [\frac{5 x^{4}}{3}] + \frac{d}{d x} [- \frac{2 x}{3}] + \frac{d}{d x} [\frac{5}{9}]$ .

$\frac{d}{d x} [\frac{5 x^{4}}{3}] + \frac{d}{d x} [- \frac{2 x}{3}] + \frac{d}{d x} [\frac{5}{9}]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{5 x^{4}}{3}]$ .

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Step 2.2.1

Since $\frac{5}{3}$ is constant with respect to $x$ , the derivative of $\frac{5 x^{4}}{3}$ with respect to $x$ is $\frac{5}{3} \frac{d}{d x} [x^{4}]$ .

$\frac{5}{3} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [- \frac{2 x}{3}] + \frac{d}{d x} [\frac{5}{9}]$

Step 2.2.2

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

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

Step 2.2.3

Combine $4$ and $\frac{5}{3}$ .

$\frac{4 \cdot 5}{3} x^{3} + \frac{d}{d x} [- \frac{2 x}{3}] + \frac{d}{d x} [\frac{5}{9}]$

Step 2.2.4

Multiply $4$ by $5$ .

$\frac{20}{3} x^{3} + \frac{d}{d x} [- \frac{2 x}{3}] + \frac{d}{d x} [\frac{5}{9}]$

Step 2.2.5

Combine $\frac{20}{3}$ and $x^{3}$ .

$\frac{20 x^{3}}{3} + \frac{d}{d x} [- \frac{2 x}{3}] + \frac{d}{d x} [\frac{5}{9}]$

Step 2.3

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

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Step 2.3.1

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

$\frac{20 x^{3}}{3} - \frac{2}{3} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{5}{9}]$

Step 2.3.2

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

$\frac{20 x^{3}}{3} - \frac{2}{3} \cdot 1 + \frac{d}{d x} [\frac{5}{9}]$

Step 2.3.3

Multiply $- 1$ by $1$ .

$\frac{20 x^{3}}{3} - \frac{2}{3} + \frac{d}{d x} [\frac{5}{9}]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $\frac{5}{9}$ is constant with respect to $x$ , the derivative of $\frac{5}{9}$ with respect to $x$ is $0$ .

$\frac{20 x^{3}}{3} - \frac{2}{3} + 0$

Step 2.4.2

Add $\frac{20 x^{3}}{3} - \frac{2}{3}$ and $0$ .

$f'' (x) = \frac{20 x^{3}}{3} - \frac{2}{3}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $\frac{20 x^{3}}{3} - \frac{2}{3}$ with respect to $x$ is $\frac{d}{d x} [\frac{20 x^{3}}{3}] + \frac{d}{d x} [- \frac{2}{3}]$ .

$\frac{d}{d x} [\frac{20 x^{3}}{3}] + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{20 x^{3}}{3}]$ .

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Step 3.2.1

Since $\frac{20}{3}$ is constant with respect to $x$ , the derivative of $\frac{20 x^{3}}{3}$ with respect to $x$ is $\frac{20}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{20}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.2

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

$\frac{20}{3} (3 x^{2}) + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.3

Combine $3$ and $\frac{20}{3}$ .

$\frac{3 \cdot 20}{3} x^{2} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.4

Multiply $3$ by $20$ .

$\frac{60}{3} x^{2} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.5

Combine $\frac{60}{3}$ and $x^{2}$ .

$\frac{60 x^{2}}{3} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.6

Cancel the common factor of $60$ and $3$ .

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Step 3.2.6.1

Factor $3$ out of $60 x^{2}$ .

$\frac{3 (20 x^{2})}{3} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.6.2

Cancel the common factors.

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Step 3.2.6.2.1

Factor $3$ out of $3$ .

$\frac{3 (20 x^{2})}{3 (1)} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.6.2.2

Cancel the common factor.

$\frac{3 (20 x^{2})}{3 \cdot 1} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.6.2.3

Rewrite the expression.

$\frac{20 x^{2}}{1} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.2.6.2.4

Divide $20 x^{2}$ by $1$ .

$20 x^{2} + \frac{d}{d x} [- \frac{2}{3}]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2}{3}$ with respect to $x$ is $0$ .

$20 x^{2} + 0$

Step 3.3.2

Add $20 x^{2}$ and $0$ .

$f''' (x) = 20 x^{2}$