Calculus Examples

$f (x) = 8 + 3 x + \frac{1}{2} x^{2} + \frac{1}{6} x^{3} + \frac{5}{24} x^{4} + \frac{1}{120} x^{5}$

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 $8 + 3 x + \frac{1}{2} x^{2} + \frac{1}{6} x^{3} + \frac{5}{24} x^{4} + \frac{1}{120} x^{5}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [3 x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$ .

$\frac{d}{d x} [8] + \frac{d}{d x} [3 x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [3 x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.2

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

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Step 1.2.1

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

$0 + 3 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.2.2

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

$0 + 3 \cdot 1 + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.2.3

Multiply $3$ by $1$ .

$0 + 3 + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.3

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

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Step 1.3.1

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

$0 + 3 + \frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.3.2

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

$0 + 3 + \frac{1}{2} (2 x) + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.3.3

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

$0 + 3 + \frac{2}{2} x + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.3.4

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

$0 + 3 + \frac{2 x}{2} + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.3.5

Cancel the common factor of $2$ .

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Step 1.3.5.1

Cancel the common factor.

$0 + 3 + \frac{2 x}{2} + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.3.5.2

Divide $x$ by $1$ .

$0 + 3 + x + \frac{d}{d x} [\frac{1}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{1}{6} x^{3}]$ .

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Step 1.4.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6} x^{3}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{3}]$ .

$0 + 3 + x + \frac{1}{6} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4.2

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

$0 + 3 + x + \frac{1}{6} (3 x^{2}) + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4.3

Combine $3$ and $\frac{1}{6}$ .

$0 + 3 + x + \frac{3}{6} x^{2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4.4

Combine $\frac{3}{6}$ and $x^{2}$ .

$0 + 3 + x + \frac{3 x^{2}}{6} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4.5

Cancel the common factor of $3$ and $6$ .

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Step 1.4.5.1

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

$0 + 3 + x + \frac{3 (x^{2})}{6} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4.5.2

Cancel the common factors.

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Step 1.4.5.2.1

Factor $3$ out of $6$ .

$0 + 3 + x + \frac{3 x^{2}}{3 \cdot 2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4.5.2.2

Cancel the common factor.

$0 + 3 + x + \frac{3 x^{2}}{3 \cdot 2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.4.5.2.3

Rewrite the expression.

$0 + 3 + x + \frac{x^{2}}{2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5

Evaluate $\frac{d}{d x} [\frac{5}{24} x^{4}]$ .

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Step 1.5.1

Since $\frac{5}{24}$ is constant with respect to $x$ , the derivative of $\frac{5}{24} x^{4}$ with respect to $x$ is $\frac{5}{24} \frac{d}{d x} [x^{4}]$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5}{24} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.2

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

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5}{24} (4 x^{3}) + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.3

Combine $4$ and $\frac{5}{24}$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{4 \cdot 5}{24} x^{3} + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.4

Multiply $4$ by $5$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{20}{24} x^{3} + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.5

Combine $\frac{20}{24}$ and $x^{3}$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{20 x^{3}}{24} + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.6

Cancel the common factor of $20$ and $24$ .

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Step 1.5.6.1

Factor $4$ out of $20 x^{3}$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{4 (5 x^{3})}{24} + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.6.2

Cancel the common factors.

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Step 1.5.6.2.1

Factor $4$ out of $24$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{4 (5 x^{3})}{4 (6)} + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.6.2.2

Cancel the common factor.

$0 + 3 + x + \frac{x^{2}}{2} + \frac{4 (5 x^{3})}{4 \cdot 6} + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.5.6.2.3

Rewrite the expression.

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{d}{d x} [\frac{1}{120} x^{5}]$

Step 1.6

Evaluate $\frac{d}{d x} [\frac{1}{120} x^{5}]$ .

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Step 1.6.1

Since $\frac{1}{120}$ is constant with respect to $x$ , the derivative of $\frac{1}{120} x^{5}$ with respect to $x$ is $\frac{1}{120} \frac{d}{d x} [x^{5}]$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{1}{120} \frac{d}{d x} [x^{5}]$

Step 1.6.2

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

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{1}{120} (5 x^{4})$

Step 1.6.3

Combine $5$ and $\frac{1}{120}$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5}{120} x^{4}$

Step 1.6.4

Combine $\frac{5}{120}$ and $x^{4}$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 x^{4}}{120}$

Step 1.6.5

Cancel the common factor of $5$ and $120$ .

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Step 1.6.5.1

Factor $5$ out of $5 x^{4}$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 (x^{4})}{120}$

Step 1.6.5.2

Cancel the common factors.

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Step 1.6.5.2.1

Factor $5$ out of $120$ .

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 x^{4}}{5 \cdot 24}$

Step 1.6.5.2.2

Cancel the common factor.

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 x^{4}}{5 \cdot 24}$

Step 1.6.5.2.3

Rewrite the expression.

$0 + 3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{x^{4}}{24}$

Step 1.7

Simplify.

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Step 1.7.1

Add $0$ and $3$ .

$3 + x + \frac{x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{x^{4}}{24}$

Step 1.7.2

Reorder terms.

$f' (x) = \frac{x^{4}}{24} + \frac{5 x^{3}}{6} + \frac{x^{2}}{2} + x + 3$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{x^{4}}{24} + \frac{5 x^{3}}{6} + \frac{x^{2}}{2} + x + 3$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{4}}{24}] + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [\frac{x^{4}}{24}] + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2

Evaluate $\frac{d}{d x} [\frac{x^{4}}{24}]$ .

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Step 2.2.1

Since $\frac{1}{24}$ is constant with respect to $x$ , the derivative of $\frac{x^{4}}{24}$ with respect to $x$ is $\frac{1}{24} \frac{d}{d x} [x^{4}]$ .

$\frac{1}{24} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.2

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

$\frac{1}{24} (4 x^{3}) + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.3

Combine $4$ and $\frac{1}{24}$ .

$\frac{4}{24} x^{3} + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.4

Combine $\frac{4}{24}$ and $x^{3}$ .

$\frac{4 x^{3}}{24} + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.5

Cancel the common factor of $4$ and $24$ .

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Step 2.2.5.1

Factor $4$ out of $4 x^{3}$ .

$\frac{4 (x^{3})}{24} + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.5.2

Cancel the common factors.

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Step 2.2.5.2.1

Factor $4$ out of $24$ .

$\frac{4 x^{3}}{4 \cdot 6} + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.5.2.2

Cancel the common factor.

$\frac{4 x^{3}}{4 \cdot 6} + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.2.5.2.3

Rewrite the expression.

$\frac{x^{3}}{6} + \frac{d}{d x} [\frac{5 x^{3}}{6}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{5 x^{3}}{6}]$ .

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Step 2.3.1

Since $\frac{5}{6}$ is constant with respect to $x$ , the derivative of $\frac{5 x^{3}}{6}$ with respect to $x$ is $\frac{5}{6} \frac{d}{d x} [x^{3}]$ .

$\frac{x^{3}}{6} + \frac{5}{6} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.2

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

$\frac{x^{3}}{6} + \frac{5}{6} (3 x^{2}) + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.3

Combine $3$ and $\frac{5}{6}$ .

$\frac{x^{3}}{6} + \frac{3 \cdot 5}{6} x^{2} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.4

Multiply $3$ by $5$ .

$\frac{x^{3}}{6} + \frac{15}{6} x^{2} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.5

Combine $\frac{15}{6}$ and $x^{2}$ .

$\frac{x^{3}}{6} + \frac{15 x^{2}}{6} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.6

Cancel the common factor of $15$ and $6$ .

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Step 2.3.6.1

Factor $3$ out of $15 x^{2}$ .

$\frac{x^{3}}{6} + \frac{3 (5 x^{2})}{6} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.6.2

Cancel the common factors.

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Step 2.3.6.2.1

Factor $3$ out of $6$ .

$\frac{x^{3}}{6} + \frac{3 (5 x^{2})}{3 (2)} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.6.2.2

Cancel the common factor.

$\frac{x^{3}}{6} + \frac{3 (5 x^{2})}{3 \cdot 2} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.3.6.2.3

Rewrite the expression.

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + \frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.4

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

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Step 2.4.1

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

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + \frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.4.2

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

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + \frac{1}{2} (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.4.3

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

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + \frac{2}{2} x + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.4.4

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

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + \frac{2 x}{2} + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.4.5

Cancel the common factor of $2$ .

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Step 2.4.5.1

Cancel the common factor.

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + \frac{2 x}{2} + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.4.5.2

Divide $x$ by $1$ .

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + x + \frac{d}{d x} [x] + \frac{d}{d x} [3]$

Step 2.5

Differentiate.

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Step 2.5.1

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

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + x + 1 + \frac{d}{d x} [3]$

Step 2.5.2

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

$\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + x + 1 + 0$

Step 2.5.3

Add $\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + x + 1$ and $0$ .

$f'' (x) = \frac{x^{3}}{6} + \frac{5 x^{2}}{2} + x + 1$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $\frac{x^{3}}{6} + \frac{5 x^{2}}{2} + x + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{3}}{6}] + \frac{d}{d x} [\frac{5 x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [\frac{x^{3}}{6}] + \frac{d}{d x} [\frac{5 x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{x^{3}}{6}]$ .

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Step 3.2.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{x^{3}}{6}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{6} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{5 x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2.2

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

$\frac{1}{6} (3 x^{2}) + \frac{d}{d x} [\frac{5 x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2.3

Combine $3$ and $\frac{1}{6}$ .

$\frac{3}{6} x^{2} + \frac{d}{d x} [\frac{5 x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2.4

Combine $\frac{3}{6}$ and $x^{2}$ .

$\frac{3 x^{2}}{6} + \frac{d}{d x} [\frac{5 x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2.5

Cancel the common factor of $3$ and $6$ .

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Step 3.2.5.1

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

$\frac{3 (x^{2})}{6} + \frac{d}{d x} [\frac{5 x^{2}}{2}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.2.5.2

Cancel the common factors.

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Step 3.2.5.2.1

Factor $3$ out of $6$ .

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

Step 3.2.5.2.2

Cancel the common factor.

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

Step 3.2.5.2.3

Rewrite the expression.

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

Step 3.3

Evaluate $\frac{d}{d x} [\frac{5 x^{2}}{2}]$ .

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Step 3.3.1

Since $\frac{5}{2}$ is constant with respect to $x$ , the derivative of $\frac{5 x^{2}}{2}$ with respect to $x$ is $\frac{5}{2} \frac{d}{d x} [x^{2}]$ .

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

Step 3.3.2

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

$\frac{x^{2}}{2} + \frac{5}{2} (2 x) + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.3.3

Combine $2$ and $\frac{5}{2}$ .

$\frac{x^{2}}{2} + \frac{2 \cdot 5}{2} x + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.3.4

Multiply $2$ by $5$ .

$\frac{x^{2}}{2} + \frac{10}{2} x + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.3.5

Combine $\frac{10}{2}$ and $x$ .

$\frac{x^{2}}{2} + \frac{10 x}{2} + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.3.6

Cancel the common factor of $10$ and $2$ .

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Step 3.3.6.1

Factor $2$ out of $10 x$ .

$\frac{x^{2}}{2} + \frac{2 (5 x)}{2} + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.3.6.2

Cancel the common factors.

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Step 3.3.6.2.1

Factor $2$ out of $2$ .

$\frac{x^{2}}{2} + \frac{2 (5 x)}{2 (1)} + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.3.6.2.2

Cancel the common factor.

$\frac{x^{2}}{2} + \frac{2 (5 x)}{2 \cdot 1} + \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 3.3.6.2.3

Rewrite the expression.

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

Step 3.3.6.2.4

Divide $5 x$ by $1$ .

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

Step 3.4

Differentiate.

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Step 3.4.1

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

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

Step 3.4.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{x^{2}}{2} + 5 x + 1 + 0$

Step 3.4.3

Add $\frac{x^{2}}{2} + 5 x + 1$ and $0$ .

$f''' (x) = \frac{x^{2}}{2} + 5 x + 1$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $\frac{x^{2}}{2} + 5 x + 1$ with respect to $x$ is $\frac{d}{d x} [\frac{x^{2}}{2}] + \frac{d}{d x} [5 x] + \frac{d}{d x} [1]$ .

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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

$\frac{1}{2} (2 x) + \frac{d}{d x} [5 x] + \frac{d}{d x} [1]$

Step 4.2.3

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

$\frac{2}{2} x + \frac{d}{d x} [5 x] + \frac{d}{d x} [1]$

Step 4.2.4

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

$\frac{2 x}{2} + \frac{d}{d x} [5 x] + \frac{d}{d x} [1]$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Cancel the common factor.

$\frac{2 x}{2} + \frac{d}{d x} [5 x] + \frac{d}{d x} [1]$

Step 4.2.5.2

Divide $x$ by $1$ .

$x + \frac{d}{d x} [5 x] + \frac{d}{d x} [1]$

Step 4.3

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

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Step 4.3.1

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

$x + 5 \frac{d}{d x} [x] + \frac{d}{d x} [1]$

Step 4.3.2

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

$x + 5 \cdot 1 + \frac{d}{d x} [1]$

Step 4.3.3

Multiply $5$ by $1$ .

$x + 5 + \frac{d}{d x} [1]$

Step 4.4

Differentiate using the Constant Rule.

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Step 4.4.1

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$x + 5 + 0$

Step 4.4.2

Add $x + 5$ and $0$ .

$f^{4} (x) = x + 5$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $x + 5$ .

$x + 5$