Calculus Examples

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

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

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

$0$ .

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