Calculus Examples

$f (x) = \ln (\frac{x^{5} - 12}{x})$

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 $g (x) = \frac{x^{5} - 12}{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{x^{5} - 12}{x}$ .

$\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{x^{5} - 12}{x}]$

Step 1.2

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

$\frac{1}{u} \frac{d}{d x} [\frac{x^{5} - 12}{x}]$

Step 1.3

Replace all occurrences of $u$ with $\frac{x^{5} - 12}{x}$ .

$\frac{1}{\frac{x^{5} - 12}{x}} \frac{d}{d x} [\frac{x^{5} - 12}{x}]$

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{x^{5} - 12}{x}$ .

$1 \frac{x}{x^{5} - 12} \frac{d}{d x} [\frac{x^{5} - 12}{x}]$

Step 3

Multiply $\frac{x}{x^{5} - 12}$ by $1$ .

$\frac{x}{x^{5} - 12} \frac{d}{d x} [\frac{x^{5} - 12}{x}]$

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 $f (x) = x^{5} - 12$ and $g (x) = x$ .

$\frac{x}{x^{5} - 12} \cdot \frac{x \frac{d}{d x} [x^{5} - 12] - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 5

Differentiate.

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Step 5.1

By the Sum Rule, the derivative of $x^{5} - 12$ with respect to $x$ is $\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 12]$ .

$\frac{x}{x^{5} - 12} \cdot \frac{x (\frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 12]) - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 5.2

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

$\frac{x}{x^{5} - 12} \cdot \frac{x (5 x^{4} + \frac{d}{d x} [- 12]) - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 5.3

Since $- 12$ is constant with respect to $x$ , the derivative of $- 12$ with respect to $x$ is $0$ .

$\frac{x}{x^{5} - 12} \cdot \frac{x (5 x^{4} + 0) - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 5.4

Add $5 x^{4}$ and $0$ .

$\frac{x}{x^{5} - 12} \cdot \frac{x (5 x^{4}) - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 6

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

$\frac{x}{x^{5} - 12} \cdot \frac{5 (x^{1} x^{4}) - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 7

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

$\frac{x}{x^{5} - 12} \cdot \frac{5 x^{1 + 4} - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 8

Add $1$ and $4$ .

$\frac{x}{x^{5} - 12} \cdot \frac{5 x^{5} - (x^{5} - 12) \frac{d}{d x} [x]}{x^{2}}$

Step 9

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

$\frac{x}{x^{5} - 12} \cdot \frac{5 x^{5} - (x^{5} - 12) \cdot 1}{x^{2}}$

Step 10

Combine fractions.

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Step 10.1

Multiply $- 1$ by $1$ .

$\frac{x}{x^{5} - 12} \cdot \frac{5 x^{5} - (x^{5} - 12)}{x^{2}}$

Step 10.2

Multiply $\frac{x}{x^{5} - 12}$ by $\frac{5 x^{5} - (x^{5} - 12)}{x^{2}}$ .

$\frac{x (5 x^{5} - (x^{5} - 12))}{(x^{5} - 12) x^{2}}$

Step 11

Cancel the common factors.

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Step 11.1

Factor $x$ out of $(x^{5} - 12) x^{2}$ .

$\frac{x (5 x^{5} - (x^{5} - 12))}{x ((x^{5} - 12) x)}$

Step 11.2

Cancel the common factor.

$\frac{x (5 x^{5} - (x^{5} - 12))}{x ((x^{5} - 12) x)}$

Step 11.3

Rewrite the expression.

$\frac{5 x^{5} - (x^{5} - 12)}{(x^{5} - 12) x}$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$\frac{5 x^{5} - x^{5} - - 12}{(x^{5} - 12) x}$

Step 12.2

Apply the distributive property.

$\frac{5 x^{5} - x^{5} - - 12}{x^{5} x - 12 x}$

Step 12.3

Simplify the numerator.

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Step 12.3.1

Multiply $- 1$ by $- 12$ .

$\frac{5 x^{5} - x^{5} + 12}{x^{5} x - 12 x}$

Step 12.3.2

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

$\frac{4 x^{5} + 12}{x^{5} x - 12 x}$

Step 12.4

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 12.4.1

Multiply $x^{5}$ by $x$ .

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Step 12.4.1.1

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

$\frac{4 x^{5} + 12}{x^{5} x^{1} - 12 x}$

Step 12.4.1.2

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

$\frac{4 x^{5} + 12}{x^{5 + 1} - 12 x}$

Step 12.4.2

Add $5$ and $1$ .

$\frac{4 x^{5} + 12}{x^{6} - 12 x}$

Step 12.5

Factor $4$ out of $4 x^{5} + 12$ .

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Step 12.5.1

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

$\frac{4 (x^{5}) + 12}{x^{6} - 12 x}$

Step 12.5.2

Factor $4$ out of $12$ .

$\frac{4 x^{5} + 4 \cdot 3}{x^{6} - 12 x}$

Step 12.5.3

Factor $4$ out of $4 x^{5} + 4 \cdot 3$ .

$\frac{4 (x^{5} + 3)}{x^{6} - 12 x}$

Step 12.6

Factor $x$ out of $x^{6} - 12 x$ .

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Step 12.6.1

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

$\frac{4 (x^{5} + 3)}{x \cdot x^{5} - 12 x}$

Step 12.6.2

Factor $x$ out of $- 12 x$ .

$\frac{4 (x^{5} + 3)}{x \cdot x^{5} + x \cdot - 12}$

Step 12.6.3

Factor $x$ out of $x \cdot x^{5} + x \cdot - 12$ .

$\frac{4 (x^{5} + 3)}{x (x^{5} - 12)}$