Calculus Examples

$f (x) = 2 \ln (\frac{11}{x})$

Step 1

Since $2$ is constant with respect to $x$ , the derivative of $2 \ln (\frac{11}{x})$ with respect to $x$ is $2 \frac{d}{d x} [\ln (\frac{11}{x})]$ .

$2 \frac{d}{d x} [\ln (\frac{11}{x})]$

Step 2

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 $g (x) = \frac{11}{x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{11}{x}$ .

$2 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{11}{x}])$

Step 2.2

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

$2 (\frac{1}{u} \frac{d}{d x} [\frac{11}{x}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{11}{x}$ .

$2 (\frac{1}{\frac{11}{x}} \frac{d}{d x} [\frac{11}{x}])$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{11}{x}$ .

$2 (1 \frac{x}{11} \frac{d}{d x} [\frac{11}{x}])$

Step 4

Combine fractions.

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Step 4.1

Multiply $\frac{x}{11}$ by $1$ .

$2 (\frac{x}{11} \frac{d}{d x} [\frac{11}{x}])$

Step 4.2

Combine $\frac{x}{11}$ and $2$ .

$\frac{x \cdot 2}{11} \frac{d}{d x} [\frac{11}{x}]$

Step 4.3

Move $2$ to the left of $x$ .

$\frac{2 \cdot x}{11} \frac{d}{d x} [\frac{11}{x}]$

Step 5

Since $11$ is constant with respect to $x$ , the derivative of $\frac{11}{x}$ with respect to $x$ is $11 \frac{d}{d x} [\frac{1}{x}]$ .

$\frac{2 x}{11} (11 \frac{d}{d x} [\frac{1}{x}])$

Step 6

Simplify terms.

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Step 6.1

Combine $11$ and $\frac{2 x}{11}$ .

$\frac{11 (2 x)}{11} \frac{d}{d x} [\frac{1}{x}]$

Step 6.2

Multiply $2$ by $11$ .

$\frac{22 x}{11} \frac{d}{d x} [\frac{1}{x}]$

Step 6.3

Cancel the common factor of $22$ and $11$ .

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Step 6.3.1

Factor $11$ out of $22 x$ .

$\frac{11 (2 x)}{11} \frac{d}{d x} [\frac{1}{x}]$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $11$ out of $11$ .

$\frac{11 (2 x)}{11 (1)} \frac{d}{d x} [\frac{1}{x}]$

Step 6.3.2.2

Cancel the common factor.

$\frac{11 (2 x)}{11 \cdot 1} \frac{d}{d x} [\frac{1}{x}]$

Step 6.3.2.3

Rewrite the expression.

$\frac{2 x}{1} \frac{d}{d x} [\frac{1}{x}]$

Step 6.3.2.4

Divide $2 x$ by $1$ .

$2 x \frac{d}{d x} [\frac{1}{x}]$

Step 6.4

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

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

Step 7

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

$2 x (- x^{- 2})$

Step 8

Multiply $- 1$ by $2$ .

$- 2 x \cdot x^{- 2}$

Step 9

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

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Step 9.1

Move $x^{- 2}$ .

$- 2 (x^{- 2} x)$

Step 9.2

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

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Step 9.2.1

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

$- 2 (x^{- 2} x^{1})$

Step 9.2.2

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

$- 2 x^{- 2 + 1}$

Step 9.3

Add $- 2$ and $1$ .

$- 2 x^{- 1}$

Step 10

Simplify.

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Step 10.1

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

$- 2 \frac{1}{x}$

Step 10.2

Combine terms.

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Step 10.2.1

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

$\frac{- 2}{x}$

Step 10.2.2

Move the negative in front of the fraction.

$- \frac{2}{x}$