Calculus Examples

$f (x) = \ln (\frac{x^{2} - 7}{x})$

Step 1

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 $g (x) = \frac{x^{2} - 7}{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x^{2} - 7}{x}$ .

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

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [\frac{x^{2} - 7}{x}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x^{2} - 7}{x}$ .

$\frac{1}{\frac{x^{2} - 7}{x}} \frac{d}{d x} [\frac{x^{2} - 7}{x}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{x^{2} - 7}{x}$ .

$1 \frac{x}{x^{2} - 7} \frac{d}{d x} [\frac{x^{2} - 7}{x}]$

Step 3

Multiply $\frac{x}{x^{2} - 7}$ by $1$ .

$\frac{x}{x^{2} - 7} \frac{d}{d x} [\frac{x^{2} - 7}{x}]$

Step 4

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 $f (x) = x^{2} - 7$ and $g (x) = x$ .

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

Step 5

Differentiate.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7$ with respect to $x$ is $0$ .

$\frac{x}{x^{2} - 7} \cdot \frac{x (2 x + 0) - (x^{2} - 7) \frac{d}{d x} [x]}{x^{2}}$

Step 5.4

Add $2 x$ and $0$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Add $1$ and $1$ .

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

Step 10

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

$\frac{x}{x^{2} - 7} \cdot \frac{2 x^{2} - (x^{2} - 7) \cdot 1}{x^{2}}$

Step 11

Combine fractions.

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Step 11.1

Multiply $- 1$ by $1$ .

$\frac{x}{x^{2} - 7} \cdot \frac{2 x^{2} - (x^{2} - 7)}{x^{2}}$

Step 11.2

Multiply $\frac{x}{x^{2} - 7}$ by $\frac{2 x^{2} - (x^{2} - 7)}{x^{2}}$ .

$\frac{x (2 x^{2} - (x^{2} - 7))}{(x^{2} - 7) x^{2}}$

Step 12

Cancel the common factors.

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Step 12.1

Factor $x$ out of $(x^{2} - 7) x^{2}$ .

$\frac{x (2 x^{2} - (x^{2} - 7))}{x ((x^{2} - 7) x)}$

Step 12.2

Cancel the common factor.

$\frac{x (2 x^{2} - (x^{2} - 7))}{x ((x^{2} - 7) x)}$

Step 12.3

Rewrite the expression.

$\frac{2 x^{2} - (x^{2} - 7)}{(x^{2} - 7) x}$

Step 13

Simplify.

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Step 13.1

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - - 7}{(x^{2} - 7) x}$

Step 13.2

Apply the distributive property.

$\frac{2 x^{2} - x^{2} - - 7}{x^{2} x - 7 x}$

Step 13.3

Simplify the numerator.

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Step 13.3.1

Multiply $- 1$ by $- 7$ .

$\frac{2 x^{2} - x^{2} + 7}{x^{2} x - 7 x}$

Step 13.3.2

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

$\frac{x^{2} + 7}{x^{2} x - 7 x}$

Step 13.4

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

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Step 13.4.1

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

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Step 13.4.1.1

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

$\frac{x^{2} + 7}{x^{2} x^{1} - 7 x}$

Step 13.4.1.2

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

$\frac{x^{2} + 7}{x^{2 + 1} - 7 x}$

Step 13.4.2

Add $2$ and $1$ .

$\frac{x^{2} + 7}{x^{3} - 7 x}$

Step 13.5

Factor $x$ out of $x^{3} - 7 x$ .

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Step 13.5.1

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

$\frac{x^{2} + 7}{x \cdot x^{2} - 7 x}$

Step 13.5.2

Factor $x$ out of $- 7 x$ .

$\frac{x^{2} + 7}{x \cdot x^{2} + x \cdot - 7}$

Step 13.5.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 7$ .

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