Calculus Examples

$f (x) = 8 + 9 x + \frac{1}{2} x^{2} + \frac{5}{6} x^{3} + \frac{5}{24} x^{4} + \frac{7}{120} x^{5}$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $8 + 9 x + \frac{1}{2} x^{2} + \frac{5}{6} x^{3} + \frac{5}{24} x^{4} + \frac{7}{120} x^{5}$ with respect to $x$ is $\frac{d}{d x} [8] + \frac{d}{d x} [9 x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$ .

$\frac{d}{d x} [8] + \frac{d}{d x} [9 x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 1.2

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

$0 + \frac{d}{d x} [9 x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 2

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

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Step 2.1

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

$0 + 9 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 2.2

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

$0 + 9 \cdot 1 + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 2.3

Multiply $9$ by $1$ .

$0 + 9 + \frac{d}{d x} [\frac{1}{2} x^{2}] + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 3

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

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

$0 + 9 + \frac{1}{2} \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 3.2

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

$0 + 9 + \frac{1}{2} (2 x) + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 3.3

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

$0 + 9 + \frac{2}{2} x + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 3.4

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

$0 + 9 + \frac{2 x}{2} + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 3.5

Cancel the common factor of $2$ .

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Step 3.5.1

Cancel the common factor.

$0 + 9 + \frac{2 x}{2} + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 3.5.2

Divide $x$ by $1$ .

$0 + 9 + x + \frac{d}{d x} [\frac{5}{6} x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4

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

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Step 4.1

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

$0 + 9 + x + \frac{5}{6} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.2

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

$0 + 9 + x + \frac{5}{6} (3 x^{2}) + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.3

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

$0 + 9 + x + \frac{3 \cdot 5}{6} x^{2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.4

Multiply $3$ by $5$ .

$0 + 9 + x + \frac{15}{6} x^{2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.5

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

$0 + 9 + x + \frac{15 x^{2}}{6} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.6

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

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Step 4.6.1

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

$0 + 9 + x + \frac{3 (5 x^{2})}{6} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.6.2

Cancel the common factors.

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Step 4.6.2.1

Factor $3$ out of $6$ .

$0 + 9 + x + \frac{3 (5 x^{2})}{3 (2)} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.6.2.2

Cancel the common factor.

$0 + 9 + x + \frac{3 (5 x^{2})}{3 \cdot 2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 4.6.2.3

Rewrite the expression.

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{d}{d x} [\frac{5}{24} x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5

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

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

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5}{24} \frac{d}{d x} [x^{4}] + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.2

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

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5}{24} (4 x^{3}) + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.3

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

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{4 \cdot 5}{24} x^{3} + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.4

Multiply $4$ by $5$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{20}{24} x^{3} + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.5

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

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{20 x^{3}}{24} + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.6

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

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Step 5.6.1

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

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{4 (5 x^{3})}{24} + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $4$ out of $24$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{4 (5 x^{3})}{4 (6)} + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.6.2.2

Cancel the common factor.

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{4 (5 x^{3})}{4 \cdot 6} + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 5.6.2.3

Rewrite the expression.

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{d}{d x} [\frac{7}{120} x^{5}]$

Step 6

Evaluate $\frac{d}{d x} [\frac{7}{120} x^{5}]$ .

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Step 6.1

Since $\frac{7}{120}$ is constant with respect to $x$ , the derivative of $\frac{7}{120} x^{5}$ with respect to $x$ is $\frac{7}{120} \frac{d}{d x} [x^{5}]$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{7}{120} \frac{d}{d x} [x^{5}]$

Step 6.2

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

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{7}{120} (5 x^{4})$

Step 6.3

Combine $5$ and $\frac{7}{120}$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 \cdot 7}{120} x^{4}$

Step 6.4

Multiply $5$ by $7$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{35}{120} x^{4}$

Step 6.5

Combine $\frac{35}{120}$ and $x^{4}$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{35 x^{4}}{120}$

Step 6.6

Cancel the common factor of $35$ and $120$ .

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Step 6.6.1

Factor $5$ out of $35 x^{4}$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 (7 x^{4})}{120}$

Step 6.6.2

Cancel the common factors.

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Step 6.6.2.1

Factor $5$ out of $120$ .

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 (7 x^{4})}{5 (24)}$

Step 6.6.2.2

Cancel the common factor.

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{5 (7 x^{4})}{5 \cdot 24}$

Step 6.6.2.3

Rewrite the expression.

$0 + 9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{7 x^{4}}{24}$

Step 7

Simplify.

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Step 7.1

Add $0$ and $9$ .

$9 + x + \frac{5 x^{2}}{2} + \frac{5 x^{3}}{6} + \frac{7 x^{4}}{24}$

Step 7.2

Reorder terms.

$\frac{7 x^{4}}{24} + \frac{5 x^{3}}{6} + \frac{5 x^{2}}{2} + x + 9$