Calculus Examples

$\frac{d^{9}}{d x^{9}} \cdot (x^{8} \ln (x))$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{8}$ and $g (x) = \ln (x)$ .

$x^{8} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$x^{8} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3

Differentiate using the Power Rule.

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Step 1.3.1

Combine $x^{8}$ and $\frac{1}{x}$ .

$\frac{x^{8}}{x} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3.2

Cancel the common factor of $x^{8}$ and $x$ .

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Step 1.3.2.1

Factor $x$ out of $x^{8}$ .

$\frac{x \cdot x^{7}}{x} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3.2.2

Cancel the common factors.

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Step 1.3.2.2.1

Raise $x$ to the power of $1$ .

$\frac{x \cdot x^{7}}{x^{1}} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3.2.2.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x^{7}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3.2.2.3

Cancel the common factor.

$\frac{x \cdot x^{7}}{x \cdot 1} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3.2.2.4

Rewrite the expression.

$\frac{x^{7}}{1} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3.2.2.5

Divide $x^{7}$ by $1$ .

$x^{7} + \ln (x) \frac{d}{d x} [x^{8}]$

Step 1.3.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 8$ .

$x^{7} + \ln (x) (8 x^{7})$

Step 1.3.4

Reorder terms.

$f' (x) = x^{7} + 8 x^{7} \ln (x)$

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

By the Sum Rule, the derivative of $x^{7} + 8 x^{7} \ln (x)$ with respect to $x$ is $\frac{d}{d x} [x^{7}] + \frac{d}{d x} [8 x^{7} \ln (x)]$ .

$\frac{d}{d x} [x^{7}] + \frac{d}{d x} [8 x^{7} \ln (x)]$

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$7 x^{6} + \frac{d}{d x} [8 x^{7} \ln (x)]$

Step 2.2

Evaluate $\frac{d}{d x} [8 x^{7} \ln (x)]$ .

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Step 2.2.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x^{7} \ln (x)$ with respect to $x$ is $8 \frac{d}{d x} [x^{7} \ln (x)]$ .

$7 x^{6} + 8 \frac{d}{d x} [x^{7} \ln (x)]$

Step 2.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{7}$ and $g (x) = \ln (x)$ .

$7 x^{6} + 8 (x^{7} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{7}])$

Step 2.2.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$7 x^{6} + 8 (x^{7} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{7}])$

Step 2.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 7$ .

$7 x^{6} + 8 (x^{7} \frac{1}{x} + \ln (x) (7 x^{6}))$

Step 2.2.5

Combine $x^{7}$ and $\frac{1}{x}$ .

$7 x^{6} + 8 (\frac{x^{7}}{x} + \ln (x) (7 x^{6}))$

Step 2.2.6

Cancel the common factor of $x^{7}$ and $x$ .

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Step 2.2.6.1

Factor $x$ out of $x^{7}$ .

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x} + \ln (x) (7 x^{6}))$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Raise $x$ to the power of $1$ .

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x^{1}} + \ln (x) (7 x^{6}))$

Step 2.2.6.2.2

Factor $x$ out of $x^{1}$ .

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6}))$

Step 2.2.6.2.3

Cancel the common factor.

$7 x^{6} + 8 (\frac{x \cdot x^{6}}{x \cdot 1} + \ln (x) (7 x^{6}))$

Step 2.2.6.2.4

Rewrite the expression.

$7 x^{6} + 8 (\frac{x^{6}}{1} + \ln (x) (7 x^{6}))$

Step 2.2.6.2.5

Divide $x^{6}$ by $1$ .

$7 x^{6} + 8 (x^{6} + \ln (x) (7 x^{6}))$

Step 2.3

Simplify.

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Step 2.3.1

Apply the distributive property.

$7 x^{6} + 8 x^{6} + 8 (\ln (x) (7 x^{6}))$

Step 2.3.2

Combine terms.

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Step 2.3.2.1

Multiply $7$ by $8$ .

$7 x^{6} + 8 x^{6} + 56 (\ln (x) (x^{6}))$

Step 2.3.2.2

Add $7 x^{6}$ and $8 x^{6}$ .

$15 x^{6} + 56 \ln (x) x^{6}$

Step 2.3.3

Reorder terms.

$f'' (x) = 15 x^{6} + 56 x^{6} \ln (x)$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $15 x^{6} + 56 x^{6} \ln (x)$ with respect to $x$ is $\frac{d}{d x} [15 x^{6}] + \frac{d}{d x} [56 x^{6} \ln (x)]$ .

$\frac{d}{d x} [15 x^{6}] + \frac{d}{d x} [56 x^{6} \ln (x)]$

Step 3.2

Evaluate $\frac{d}{d x} [15 x^{6}]$ .

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Step 3.2.1

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{6}$ with respect to $x$ is $15 \frac{d}{d x} [x^{6}]$ .

$15 \frac{d}{d x} [x^{6}] + \frac{d}{d x} [56 x^{6} \ln (x)]$

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$15 (6 x^{5}) + \frac{d}{d x} [56 x^{6} \ln (x)]$

Step 3.2.3

Multiply $6$ by $15$ .

$90 x^{5} + \frac{d}{d x} [56 x^{6} \ln (x)]$

Step 3.3

Evaluate $\frac{d}{d x} [56 x^{6} \ln (x)]$ .

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Step 3.3.1

Since $56$ is constant with respect to $x$ , the derivative of $56 x^{6} \ln (x)$ with respect to $x$ is $56 \frac{d}{d x} [x^{6} \ln (x)]$ .

$90 x^{5} + 56 \frac{d}{d x} [x^{6} \ln (x)]$

Step 3.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{6}$ and $g (x) = \ln (x)$ .

$90 x^{5} + 56 (x^{6} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{6}])$

Step 3.3.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$90 x^{5} + 56 (x^{6} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{6}])$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$90 x^{5} + 56 (x^{6} \frac{1}{x} + \ln (x) (6 x^{5}))$

Step 3.3.5

Combine $x^{6}$ and $\frac{1}{x}$ .

$90 x^{5} + 56 (\frac{x^{6}}{x} + \ln (x) (6 x^{5}))$

Step 3.3.6

Cancel the common factor of $x^{6}$ and $x$ .

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Step 3.3.6.1

Factor $x$ out of $x^{6}$ .

$90 x^{5} + 56 (\frac{x \cdot x^{5}}{x} + \ln (x) (6 x^{5}))$

Step 3.3.6.2

Cancel the common factors.

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Step 3.3.6.2.1

Raise $x$ to the power of $1$ .

$90 x^{5} + 56 (\frac{x \cdot x^{5}}{x^{1}} + \ln (x) (6 x^{5}))$

Step 3.3.6.2.2

Factor $x$ out of $x^{1}$ .

$90 x^{5} + 56 (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (x) (6 x^{5}))$

Step 3.3.6.2.3

Cancel the common factor.

$90 x^{5} + 56 (\frac{x \cdot x^{5}}{x \cdot 1} + \ln (x) (6 x^{5}))$

Step 3.3.6.2.4

Rewrite the expression.

$90 x^{5} + 56 (\frac{x^{5}}{1} + \ln (x) (6 x^{5}))$

Step 3.3.6.2.5

Divide $x^{5}$ by $1$ .

$90 x^{5} + 56 (x^{5} + \ln (x) (6 x^{5}))$

Step 3.4

Simplify.

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Step 3.4.1

Apply the distributive property.

$90 x^{5} + 56 x^{5} + 56 (\ln (x) (6 x^{5}))$

Step 3.4.2

Combine terms.

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Step 3.4.2.1

Multiply $6$ by $56$ .

$90 x^{5} + 56 x^{5} + 336 (\ln (x) (x^{5}))$

Step 3.4.2.2

Add $90 x^{5}$ and $56 x^{5}$ .

$146 x^{5} + 336 \ln (x) x^{5}$

Step 3.4.3

Reorder terms.

$f''' (x) = 146 x^{5} + 336 x^{5} \ln (x)$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $146 x^{5} + 336 x^{5} \ln (x)$ with respect to $x$ is $\frac{d}{d x} [146 x^{5}] + \frac{d}{d x} [336 x^{5} \ln (x)]$ .

$\frac{d}{d x} [146 x^{5}] + \frac{d}{d x} [336 x^{5} \ln (x)]$

Step 4.2

Evaluate $\frac{d}{d x} [146 x^{5}]$ .

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Step 4.2.1

Since $146$ is constant with respect to $x$ , the derivative of $146 x^{5}$ with respect to $x$ is $146 \frac{d}{d x} [x^{5}]$ .

$146 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [336 x^{5} \ln (x)]$

Step 4.2.2

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

$146 (5 x^{4}) + \frac{d}{d x} [336 x^{5} \ln (x)]$

Step 4.2.3

Multiply $5$ by $146$ .

$730 x^{4} + \frac{d}{d x} [336 x^{5} \ln (x)]$

Step 4.3

Evaluate $\frac{d}{d x} [336 x^{5} \ln (x)]$ .

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Step 4.3.1

Since $336$ is constant with respect to $x$ , the derivative of $336 x^{5} \ln (x)$ with respect to $x$ is $336 \frac{d}{d x} [x^{5} \ln (x)]$ .

$730 x^{4} + 336 \frac{d}{d x} [x^{5} \ln (x)]$

Step 4.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{5}$ and $g (x) = \ln (x)$ .

$730 x^{4} + 336 (x^{5} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.3.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$730 x^{4} + 336 (x^{5} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{5}])$

Step 4.3.4

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

$730 x^{4} + 336 (x^{5} \frac{1}{x} + \ln (x) (5 x^{4}))$

Step 4.3.5

Combine $x^{5}$ and $\frac{1}{x}$ .

$730 x^{4} + 336 (\frac{x^{5}}{x} + \ln (x) (5 x^{4}))$

Step 4.3.6

Cancel the common factor of $x^{5}$ and $x$ .

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Step 4.3.6.1

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

$730 x^{4} + 336 (\frac{x \cdot x^{4}}{x} + \ln (x) (5 x^{4}))$

Step 4.3.6.2

Cancel the common factors.

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Step 4.3.6.2.1

Raise $x$ to the power of $1$ .

$730 x^{4} + 336 (\frac{x \cdot x^{4}}{x^{1}} + \ln (x) (5 x^{4}))$

Step 4.3.6.2.2

Factor $x$ out of $x^{1}$ .

$730 x^{4} + 336 (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) (5 x^{4}))$

Step 4.3.6.2.3

Cancel the common factor.

$730 x^{4} + 336 (\frac{x \cdot x^{4}}{x \cdot 1} + \ln (x) (5 x^{4}))$

Step 4.3.6.2.4

Rewrite the expression.

$730 x^{4} + 336 (\frac{x^{4}}{1} + \ln (x) (5 x^{4}))$

Step 4.3.6.2.5

Divide $x^{4}$ by $1$ .

$730 x^{4} + 336 (x^{4} + \ln (x) (5 x^{4}))$

Step 4.4

Simplify.

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Step 4.4.1

Apply the distributive property.

$730 x^{4} + 336 x^{4} + 336 (\ln (x) (5 x^{4}))$

Step 4.4.2

Combine terms.

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Step 4.4.2.1

Multiply $5$ by $336$ .

$730 x^{4} + 336 x^{4} + 1680 (\ln (x) (x^{4}))$

Step 4.4.2.2

Add $730 x^{4}$ and $336 x^{4}$ .

$1066 x^{4} + 1680 \ln (x) x^{4}$

Step 4.4.3

Reorder terms.

$f^{4} (x) = 1066 x^{4} + 1680 x^{4} \ln (x)$

Step 5

Find the 5th derivative.

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Step 5.1

By the Sum Rule, the derivative of $1066 x^{4} + 1680 x^{4} \ln (x)$ with respect to $x$ is $\frac{d}{d x} [1066 x^{4}] + \frac{d}{d x} [1680 x^{4} \ln (x)]$ .

$\frac{d}{d x} [1066 x^{4}] + \frac{d}{d x} [1680 x^{4} \ln (x)]$

Step 5.2

Evaluate $\frac{d}{d x} [1066 x^{4}]$ .

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Step 5.2.1

Since $1066$ is constant with respect to $x$ , the derivative of $1066 x^{4}$ with respect to $x$ is $1066 \frac{d}{d x} [x^{4}]$ .

$1066 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [1680 x^{4} \ln (x)]$

Step 5.2.2

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

$1066 (4 x^{3}) + \frac{d}{d x} [1680 x^{4} \ln (x)]$

Step 5.2.3

Multiply $4$ by $1066$ .

$4264 x^{3} + \frac{d}{d x} [1680 x^{4} \ln (x)]$

Step 5.3

Evaluate $\frac{d}{d x} [1680 x^{4} \ln (x)]$ .

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Step 5.3.1

Since $1680$ is constant with respect to $x$ , the derivative of $1680 x^{4} \ln (x)$ with respect to $x$ is $1680 \frac{d}{d x} [x^{4} \ln (x)]$ .

$4264 x^{3} + 1680 \frac{d}{d x} [x^{4} \ln (x)]$

Step 5.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{4}$ and $g (x) = \ln (x)$ .

$4264 x^{3} + 1680 (x^{4} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{4}])$

Step 5.3.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$4264 x^{3} + 1680 (x^{4} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{4}])$

Step 5.3.4

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

$4264 x^{3} + 1680 (x^{4} \frac{1}{x} + \ln (x) (4 x^{3}))$

Step 5.3.5

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

$4264 x^{3} + 1680 (\frac{x^{4}}{x} + \ln (x) (4 x^{3}))$

Step 5.3.6

Cancel the common factor of $x^{4}$ and $x$ .

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Step 5.3.6.1

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

$4264 x^{3} + 1680 (\frac{x \cdot x^{3}}{x} + \ln (x) (4 x^{3}))$

Step 5.3.6.2

Cancel the common factors.

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Step 5.3.6.2.1

Raise $x$ to the power of $1$ .

$4264 x^{3} + 1680 (\frac{x \cdot x^{3}}{x^{1}} + \ln (x) (4 x^{3}))$

Step 5.3.6.2.2

Factor $x$ out of $x^{1}$ .

$4264 x^{3} + 1680 (\frac{x \cdot x^{3}}{x \cdot 1} + \ln (x) (4 x^{3}))$

Step 5.3.6.2.3

Cancel the common factor.

$4264 x^{3} + 1680 (\frac{x \cdot x^{3}}{x \cdot 1} + \ln (x) (4 x^{3}))$

Step 5.3.6.2.4

Rewrite the expression.

$4264 x^{3} + 1680 (\frac{x^{3}}{1} + \ln (x) (4 x^{3}))$

Step 5.3.6.2.5

Divide $x^{3}$ by $1$ .

$4264 x^{3} + 1680 (x^{3} + \ln (x) (4 x^{3}))$

Step 5.4

Simplify.

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Step 5.4.1

Apply the distributive property.

$4264 x^{3} + 1680 x^{3} + 1680 (\ln (x) (4 x^{3}))$

Step 5.4.2

Combine terms.

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Step 5.4.2.1

Multiply $4$ by $1680$ .

$4264 x^{3} + 1680 x^{3} + 6720 (\ln (x) (x^{3}))$

Step 5.4.2.2

Add $4264 x^{3}$ and $1680 x^{3}$ .

$5944 x^{3} + 6720 \ln (x) x^{3}$

Step 5.4.3

Reorder terms.

$f^{5} (x) = 5944 x^{3} + 6720 x^{3} \ln (x)$

Step 6

Find the 6th derivative.

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Step 6.1

By the Sum Rule, the derivative of $5944 x^{3} + 6720 x^{3} \ln (x)$ with respect to $x$ is $\frac{d}{d x} [5944 x^{3}] + \frac{d}{d x} [6720 x^{3} \ln (x)]$ .

$\frac{d}{d x} [5944 x^{3}] + \frac{d}{d x} [6720 x^{3} \ln (x)]$

Step 6.2

Evaluate $\frac{d}{d x} [5944 x^{3}]$ .

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Step 6.2.1

Since $5944$ is constant with respect to $x$ , the derivative of $5944 x^{3}$ with respect to $x$ is $5944 \frac{d}{d x} [x^{3}]$ .

$5944 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [6720 x^{3} \ln (x)]$

Step 6.2.2

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

$5944 (3 x^{2}) + \frac{d}{d x} [6720 x^{3} \ln (x)]$

Step 6.2.3

Multiply $3$ by $5944$ .

$17832 x^{2} + \frac{d}{d x} [6720 x^{3} \ln (x)]$

Step 6.3

Evaluate $\frac{d}{d x} [6720 x^{3} \ln (x)]$ .

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Step 6.3.1

Since $6720$ is constant with respect to $x$ , the derivative of $6720 x^{3} \ln (x)$ with respect to $x$ is $6720 \frac{d}{d x} [x^{3} \ln (x)]$ .

$17832 x^{2} + 6720 \frac{d}{d x} [x^{3} \ln (x)]$

Step 6.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3}$ and $g (x) = \ln (x)$ .

$17832 x^{2} + 6720 (x^{3} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{3}])$

Step 6.3.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$17832 x^{2} + 6720 (x^{3} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{3}])$

Step 6.3.4

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

$17832 x^{2} + 6720 (x^{3} \frac{1}{x} + \ln (x) (3 x^{2}))$

Step 6.3.5

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

$17832 x^{2} + 6720 (\frac{x^{3}}{x} + \ln (x) (3 x^{2}))$

Step 6.3.6

Cancel the common factor of $x^{3}$ and $x$ .

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Step 6.3.6.1

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

$17832 x^{2} + 6720 (\frac{x \cdot x^{2}}{x} + \ln (x) (3 x^{2}))$

Step 6.3.6.2

Cancel the common factors.

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Step 6.3.6.2.1

Raise $x$ to the power of $1$ .

$17832 x^{2} + 6720 (\frac{x \cdot x^{2}}{x^{1}} + \ln (x) (3 x^{2}))$

Step 6.3.6.2.2

Factor $x$ out of $x^{1}$ .

$17832 x^{2} + 6720 (\frac{x \cdot x^{2}}{x \cdot 1} + \ln (x) (3 x^{2}))$

Step 6.3.6.2.3

Cancel the common factor.

$17832 x^{2} + 6720 (\frac{x \cdot x^{2}}{x \cdot 1} + \ln (x) (3 x^{2}))$

Step 6.3.6.2.4

Rewrite the expression.

$17832 x^{2} + 6720 (\frac{x^{2}}{1} + \ln (x) (3 x^{2}))$

Step 6.3.6.2.5

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

$17832 x^{2} + 6720 (x^{2} + \ln (x) (3 x^{2}))$

Step 6.4

Simplify.

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Step 6.4.1

Apply the distributive property.

$17832 x^{2} + 6720 x^{2} + 6720 (\ln (x) (3 x^{2}))$

Step 6.4.2

Combine terms.

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Step 6.4.2.1

Multiply $3$ by $6720$ .

$17832 x^{2} + 6720 x^{2} + 20160 (\ln (x) (x^{2}))$

Step 6.4.2.2

Add $17832 x^{2}$ and $6720 x^{2}$ .

$24552 x^{2} + 20160 \ln (x) x^{2}$

Step 6.4.3

Reorder terms.

$f^{6} (x) = 24552 x^{2} + 20160 x^{2} \ln (x)$

Step 7

Find the 7th derivative.

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Step 7.1

By the Sum Rule, the derivative of $24552 x^{2} + 20160 x^{2} \ln (x)$ with respect to $x$ is $\frac{d}{d x} [24552 x^{2}] + \frac{d}{d x} [20160 x^{2} \ln (x)]$ .

$\frac{d}{d x} [24552 x^{2}] + \frac{d}{d x} [20160 x^{2} \ln (x)]$

Step 7.2

Evaluate $\frac{d}{d x} [24552 x^{2}]$ .

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Step 7.2.1

Since $24552$ is constant with respect to $x$ , the derivative of $24552 x^{2}$ with respect to $x$ is $24552 \frac{d}{d x} [x^{2}]$ .

$24552 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [20160 x^{2} \ln (x)]$

Step 7.2.2

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

$24552 (2 x) + \frac{d}{d x} [20160 x^{2} \ln (x)]$

Step 7.2.3

Multiply $2$ by $24552$ .

$49104 x + \frac{d}{d x} [20160 x^{2} \ln (x)]$

Step 7.3

Evaluate $\frac{d}{d x} [20160 x^{2} \ln (x)]$ .

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Step 7.3.1

Since $20160$ is constant with respect to $x$ , the derivative of $20160 x^{2} \ln (x)$ with respect to $x$ is $20160 \frac{d}{d x} [x^{2} \ln (x)]$ .

$49104 x + 20160 \frac{d}{d x} [x^{2} \ln (x)]$

Step 7.3.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{2}$ and $g (x) = \ln (x)$ .

$49104 x + 20160 (x^{2} \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x^{2}])$

Step 7.3.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$49104 x + 20160 (x^{2} \frac{1}{x} + \ln (x) \frac{d}{d x} [x^{2}])$

Step 7.3.4

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

$49104 x + 20160 (x^{2} \frac{1}{x} + \ln (x) (2 x))$

Step 7.3.5

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

$49104 x + 20160 (\frac{x^{2}}{x} + \ln (x) (2 x))$

Step 7.3.6

Cancel the common factor of $x^{2}$ and $x$ .

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Step 7.3.6.1

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

$49104 x + 20160 (\frac{x \cdot x}{x} + \ln (x) (2 x))$

Step 7.3.6.2

Cancel the common factors.

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Step 7.3.6.2.1

Raise $x$ to the power of $1$ .

$49104 x + 20160 (\frac{x \cdot x}{x^{1}} + \ln (x) (2 x))$

Step 7.3.6.2.2

Factor $x$ out of $x^{1}$ .

$49104 x + 20160 (\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x))$

Step 7.3.6.2.3

Cancel the common factor.

$49104 x + 20160 (\frac{x \cdot x}{x \cdot 1} + \ln (x) (2 x))$

Step 7.3.6.2.4

Rewrite the expression.

$49104 x + 20160 (\frac{x}{1} + \ln (x) (2 x))$

Step 7.3.6.2.5

Divide $x$ by $1$ .

$49104 x + 20160 (x + \ln (x) (2 x))$

Step 7.4

Simplify.

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Step 7.4.1

Apply the distributive property.

$49104 x + 20160 x + 20160 (\ln (x) (2 x))$

Step 7.4.2

Combine terms.

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Step 7.4.2.1

Multiply $2$ by $20160$ .

$49104 x + 20160 x + 40320 (\ln (x) (x))$

Step 7.4.2.2

Add $49104 x$ and $20160 x$ .

$69264 x + 40320 \ln (x) x$

Step 7.4.3

Reorder terms.

$f^{7} (x) = 40320 x \ln (x) + 69264 x$

Step 8

Find the 8th derivative.

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Step 8.1

By the Sum Rule, the derivative of $40320 x \ln (x) + 69264 x$ with respect to $x$ is $\frac{d}{d x} [40320 x \ln (x)] + \frac{d}{d x} [69264 x]$ .

$\frac{d}{d x} [40320 x \ln (x)] + \frac{d}{d x} [69264 x]$

Step 8.2

Evaluate $\frac{d}{d x} [40320 x \ln (x)]$ .

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Step 8.2.1

Since $40320$ is constant with respect to $x$ , the derivative of $40320 x \ln (x)$ with respect to $x$ is $40320 \frac{d}{d x} [x \ln (x)]$ .

$40320 \frac{d}{d x} [x \ln (x)] + \frac{d}{d x} [69264 x]$

Step 8.2.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

$40320 (x \frac{d}{d x} [\ln (x)] + \ln (x) \frac{d}{d x} [x]) + \frac{d}{d x} [69264 x]$

Step 8.2.3

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$40320 (x \frac{1}{x} + \ln (x) \frac{d}{d x} [x]) + \frac{d}{d x} [69264 x]$

Step 8.2.4

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

$40320 (x \frac{1}{x} + \ln (x) \cdot 1) + \frac{d}{d x} [69264 x]$

Step 8.2.5

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

$40320 (\frac{x}{x} + \ln (x) \cdot 1) + \frac{d}{d x} [69264 x]$

Step 8.2.6

Cancel the common factor of $x$ .

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Step 8.2.6.1

Cancel the common factor.

$40320 (\frac{x}{x} + \ln (x) \cdot 1) + \frac{d}{d x} [69264 x]$

Step 8.2.6.2

Rewrite the expression.

$40320 (1 + \ln (x) \cdot 1) + \frac{d}{d x} [69264 x]$

Step 8.2.7

Multiply $\ln (x)$ by $1$ .

$40320 (1 + \ln (x)) + \frac{d}{d x} [69264 x]$

Step 8.3

Evaluate $\frac{d}{d x} [69264 x]$ .

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Step 8.3.1

Since $69264$ is constant with respect to $x$ , the derivative of $69264 x$ with respect to $x$ is $69264 \frac{d}{d x} [x]$ .

$40320 (1 + \ln (x)) + 69264 \frac{d}{d x} [x]$

Step 8.3.2

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

$40320 (1 + \ln (x)) + 69264 \cdot 1$

Step 8.3.3

Multiply $69264$ by $1$ .

$40320 (1 + \ln (x)) + 69264$

Step 8.4

Simplify.

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Step 8.4.1

Apply the distributive property.

$40320 \cdot 1 + 40320 \ln (x) + 69264$

Step 8.4.2

Combine terms.

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Step 8.4.2.1

Multiply $40320$ by $1$ .

$40320 + 40320 \ln (x) + 69264$

Step 8.4.2.2

Add $40320$ and $69264$ .

$f^{8} (x) = 40320 \ln (x) + 109584$

Step 9

Find the 9th derivative.

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Step 9.1

By the Sum Rule, the derivative of $40320 \ln (x) + 109584$ with respect to $x$ is $\frac{d}{d x} [40320 \ln (x)] + \frac{d}{d x} [109584]$ .

$\frac{d}{d x} [40320 \ln (x)] + \frac{d}{d x} [109584]$

Step 9.2

Evaluate $\frac{d}{d x} [40320 \ln (x)]$ .

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Step 9.2.1

Since $40320$ is constant with respect to $x$ , the derivative of $40320 \ln (x)$ with respect to $x$ is $40320 \frac{d}{d x} [\ln (x)]$ .

$40320 \frac{d}{d x} [\ln (x)] + \frac{d}{d x} [109584]$

Step 9.2.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$40320 \frac{1}{x} + \frac{d}{d x} [109584]$

Step 9.2.3

Combine $40320$ and $\frac{1}{x}$ .

$\frac{40320}{x} + \frac{d}{d x} [109584]$

Step 9.3

Differentiate using the Constant Rule.

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Step 9.3.1

Since $109584$ is constant with respect to $x$ , the derivative of $109584$ with respect to $x$ is $0$ .

$\frac{40320}{x} + 0$

Step 9.3.2

Add $\frac{40320}{x}$ and $0$ .

$f^{9} (x) = \frac{40320}{x}$