Calculus Examples

$\frac{x^{2} - \ln (\frac{10}{x})}{7 x^{2} + 5 x}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x^{2} - \ln (\frac{10}{x})$ and $g (x) = 7 x^{2} + 5 x$ .

$\frac{(7 x^{2} + 5 x) \frac{d}{d x} [x^{2} - \ln (\frac{10}{x})] - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - \ln (\frac{10}{x})$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \ln (\frac{10}{x})]$ .

$\frac{(7 x^{2} + 5 x) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \ln (\frac{10}{x})]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 2.2

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

$\frac{(7 x^{2} + 5 x) (2 x + \frac{d}{d x} [- \ln (\frac{10}{x})]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- \ln (\frac{10}{x})$ with respect to $x$ is $- \frac{d}{d x} [\ln (\frac{10}{x})]$ .

$\frac{(7 x^{2} + 5 x) (2 x - \frac{d}{d x} [\ln (\frac{10}{x})]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{10}{x}$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $\frac{10}{x}$ .

$\frac{(7 x^{2} + 5 x) (2 x - (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{10}{x}])) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 3.2

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

$\frac{(7 x^{2} + 5 x) (2 x - (\frac{1}{u} \frac{d}{d x} [\frac{10}{x}])) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 3.3

Replace all occurrences of $u$ with $\frac{10}{x}$ .

$\frac{(7 x^{2} + 5 x) (2 x - (\frac{1}{\frac{10}{x}} \frac{d}{d x} [\frac{10}{x}])) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{10}{x}$ .

$\frac{(7 x^{2} + 5 x) (2 x - (1 \frac{x}{10} \frac{d}{d x} [\frac{10}{x}])) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 5

Multiply $\frac{x}{10}$ by $1$ .

$\frac{(7 x^{2} + 5 x) (2 x - (\frac{x}{10} \frac{d}{d x} [\frac{10}{x}])) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 6

Since $10$ is constant with respect to $x$ , the derivative of $\frac{10}{x}$ with respect to $x$ is $10 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{(7 x^{2} + 5 x) (2 x - \frac{x}{10} (10 \frac{d}{d x} [\frac{1}{x}])) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7

Simplify terms.

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Step 7.1

Multiply $10$ by $- 1$ .

$\frac{(7 x^{2} + 5 x) (2 x - 10 \frac{x}{10} \frac{d}{d x} [\frac{1}{x}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7.2

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

$\frac{(7 x^{2} + 5 x) (2 x + \frac{- 10 x}{10} \frac{d}{d x} [\frac{1}{x}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7.3

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

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Step 7.3.1

Factor $10$ out of $- 10 x$ .

$\frac{(7 x^{2} + 5 x) (2 x + \frac{10 (- x)}{10} \frac{d}{d x} [\frac{1}{x}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7.3.2

Cancel the common factors.

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Step 7.3.2.1

Factor $10$ out of $10$ .

$\frac{(7 x^{2} + 5 x) (2 x + \frac{10 (- x)}{10 (1)} \frac{d}{d x} [\frac{1}{x}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7.3.2.2

Cancel the common factor.

$\frac{(7 x^{2} + 5 x) (2 x + \frac{10 (- x)}{10 \cdot 1} \frac{d}{d x} [\frac{1}{x}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7.3.2.3

Rewrite the expression.

$\frac{(7 x^{2} + 5 x) (2 x + \frac{- x}{1} \frac{d}{d x} [\frac{1}{x}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7.3.2.4

Divide $- x$ by $1$ .

$\frac{(7 x^{2} + 5 x) (2 x - x \frac{d}{d x} [\frac{1}{x}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 7.4

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

$\frac{(7 x^{2} + 5 x) (2 x - x \frac{d}{d x} [x^{- 1}]) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 8

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

$\frac{(7 x^{2} + 5 x) (2 x - x (- x^{- 2})) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 9

Multiply.

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Step 9.1

Multiply $- 1$ by $- 1$ .

$\frac{(7 x^{2} + 5 x) (2 x + 1 x \cdot x^{- 2}) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 9.2

Multiply $x$ by $1$ .

$\frac{(7 x^{2} + 5 x) (2 x + x \cdot x^{- 2}) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 10

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 10.1

Multiply $x$ by $x^{- 2}$ .

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Step 10.1.1

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

$\frac{(7 x^{2} + 5 x) (2 x + x^{1} x^{- 2}) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 10.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(7 x^{2} + 5 x) (2 x + x^{1 - 2}) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 10.2

Subtract $2$ from $1$ .

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) \frac{d}{d x} [7 x^{2} + 5 x]}{{(7 x^{2} + 5 x)}^{2}}$

Step 11

By the Sum Rule, the derivative of $7 x^{2} + 5 x$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [5 x]$ .

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) (\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [5 x])}{{(7 x^{2} + 5 x)}^{2}}$

Step 12

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) (7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [5 x])}{{(7 x^{2} + 5 x)}^{2}}$

Step 13

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

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) (7 (2 x) + \frac{d}{d x} [5 x])}{{(7 x^{2} + 5 x)}^{2}}$

Step 14

Multiply $2$ by $7$ .

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) (14 x + \frac{d}{d x} [5 x])}{{(7 x^{2} + 5 x)}^{2}}$

Step 15

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

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) (14 x + 5 \frac{d}{d x} [x])}{{(7 x^{2} + 5 x)}^{2}}$

Step 16

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

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) (14 x + 5 \cdot 1)}{{(7 x^{2} + 5 x)}^{2}}$

Step 17

Multiply $5$ by $1$ .

$\frac{(7 x^{2} + 5 x) (2 x + x^{- 1}) - (x^{2} - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 18

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{(7 x^{2} + 5 x) (2 x + \frac{1}{x}) - (x^{2} - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19

Simplify.

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Step 19.1

Apply the distributive property.

$\frac{(7 x^{2} + 5 x) (2 x + \frac{1}{x}) + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2

Simplify the numerator.

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Step 19.2.1

Simplify each term.

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Step 19.2.1.1

Expand $(7 x^{2} + 5 x) (2 x + \frac{1}{x})$ using the FOIL Method.

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Step 19.2.1.1.1

Apply the distributive property.

$\frac{7 x^{2} (2 x + \frac{1}{x}) + 5 x (2 x + \frac{1}{x}) + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.1.2

Apply the distributive property.

$\frac{7 x^{2} (2 x) + 7 x^{2} \frac{1}{x} + 5 x (2 x + \frac{1}{x}) + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.1.3

Apply the distributive property.

$\frac{7 x^{2} (2 x) + 7 x^{2} \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2

Simplify each term.

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Step 19.2.1.2.1

Rewrite using the commutative property of multiplication.

$\frac{7 \cdot 2 x^{2} x + 7 x^{2} \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 19.2.1.2.2.1

Move $x$ .

$\frac{7 \cdot 2 (x \cdot x^{2}) + 7 x^{2} \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.2.2

Multiply $x$ by $x^{2}$ .

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Step 19.2.1.2.2.2.1

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

$\frac{7 \cdot 2 (x^{1} x^{2}) + 7 x^{2} \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{7 \cdot 2 x^{1 + 2} + 7 x^{2} \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.2.3

Add $1$ and $2$ .

$\frac{7 \cdot 2 x^{3} + 7 x^{2} \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.3

Multiply $7$ by $2$ .

$\frac{14 x^{3} + 7 x^{2} \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.4

Cancel the common factor of $x$ .

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Step 19.2.1.2.4.1

Factor $x$ out of $7 x^{2}$ .

$\frac{14 x^{3} + x (7 x) \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.4.2

Cancel the common factor.

$\frac{14 x^{3} + x (7 x) \frac{1}{x} + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.4.3

Rewrite the expression.

$\frac{14 x^{3} + 7 x + 5 x (2 x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.5

Rewrite using the commutative property of multiplication.

$\frac{14 x^{3} + 7 x + 5 \cdot 2 x \cdot x + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.6

Multiply $x$ by $x$ by adding the exponents.

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Step 19.2.1.2.6.1

Move $x$ .

$\frac{14 x^{3} + 7 x + 5 \cdot 2 (x \cdot x) + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.6.2

Multiply $x$ by $x$ .

$\frac{14 x^{3} + 7 x + 5 \cdot 2 x^{2} + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.7

Multiply $5$ by $2$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 x \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.8

Cancel the common factor of $x$ .

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Step 19.2.1.2.8.1

Factor $x$ out of $5 x$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + x \cdot 5 \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.8.2

Cancel the common factor.

$\frac{14 x^{3} + 7 x + 10 x^{2} + x \cdot 5 \frac{1}{x} + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.2.8.3

Rewrite the expression.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 + (- x^{2} - - \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.3

Multiply $- - \ln (\frac{10}{x})$ .

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Step 19.2.1.3.1

Multiply $- 1$ by $- 1$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 + (- x^{2} + 1 \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.3.2

Multiply $\ln (\frac{10}{x})$ by $1$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 + (- x^{2} + \ln (\frac{10}{x})) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.4

Expand $(- x^{2} + \ln (\frac{10}{x})) (14 x + 5)$ using the FOIL Method.

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Step 19.2.1.4.1

Apply the distributive property.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - x^{2} (14 x + 5) + \ln (\frac{10}{x}) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.4.2

Apply the distributive property.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - x^{2} (14 x) - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x + 5)}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.4.3

Apply the distributive property.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - x^{2} (14 x) - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5

Simplify each term.

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Step 19.2.1.5.1

Rewrite using the commutative property of multiplication.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 1 \cdot 14 x^{2} x - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 19.2.1.5.2.1

Move $x$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 1 \cdot 14 (x \cdot x^{2}) - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.2.2

Multiply $x$ by $x^{2}$ .

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Step 19.2.1.5.2.2.1

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

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 1 \cdot 14 (x^{1} x^{2}) - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 1 \cdot 14 x^{1 + 2} - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.2.3

Add $1$ and $2$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 1 \cdot 14 x^{3} - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.3

Multiply $- 1$ by $14$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - x^{2} \cdot 5 + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.4

Multiply $5$ by $- 1$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{10}{x}) (14 x) + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.5

Rewrite using the commutative property of multiplication.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + 14 \ln (\frac{10}{x}) x + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.6

Simplify $14 \ln (\frac{10}{x})$ by moving $14$ inside the logarithm.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln ({(\frac{10}{x})}^{14}) x + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.7

Apply the product rule to $\frac{10}{x}$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{10^{14}}{x^{14}}) x + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.8

Raise $10$ to the power of $14$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{10}{x}) \cdot 5}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.9

Multiply $\ln (\frac{10}{x}) \cdot 5$ .

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Step 19.2.1.5.9.1

Reorder $\ln (\frac{10}{x})$ and $5$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + 5 \cdot \ln (\frac{10}{x})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.9.2

Simplify $5 \cdot \ln (\frac{10}{x})$ by moving $5$ inside the logarithm.

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + \ln ({(\frac{10}{x})}^{5})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.10

Apply the product rule to $\frac{10}{x}$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{10^{5}}{x^{5}})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.1.5.11

Raise $10$ to the power of $5$ .

$\frac{14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{100000}{x^{5}})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.2

Combine the opposite terms in $14 x^{3} + 7 x + 10 x^{2} + 5 - 14 x^{3} - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{100000}{x^{5}})$ .

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Step 19.2.2.1

Subtract $14 x^{3}$ from $14 x^{3}$ .

$\frac{7 x + 10 x^{2} + 5 + 0 - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{100000}{x^{5}})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.2.2

Add $7 x + 10 x^{2} + 5$ and $0$ .

$\frac{7 x + 10 x^{2} + 5 - 5 x^{2} + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{100000}{x^{5}})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.3

Subtract $5 x^{2}$ from $10 x^{2}$ .

$\frac{7 x + 5 x^{2} + 5 + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{100000}{x^{5}})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.2.4

Reorder factors in $7 x + 5 x^{2} + 5 + \ln (\frac{100000000000000}{x^{14}}) x + \ln (\frac{100000}{x^{5}})$ .

$\frac{7 x + 5 x^{2} + 5 + x \ln (\frac{100000000000000}{x^{14}}) + \ln (\frac{100000}{x^{5}})}{{(7 x^{2} + 5 x)}^{2}}$

Step 19.3

Reorder terms.

$\frac{5 x^{2} + x \ln (\frac{100000000000000}{x^{14}}) + 7 x + 5 + \ln (\frac{100000}{x^{5}})}{{(7 x^{2} + 5 x)}^{2}}$