Calculus Examples

$3 \ln (\frac{1}{x})$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 \ln (\frac{1}{x})$ with respect to $x$ is $3 \frac{d}{d x} [\ln (\frac{1}{x})]$ .

$3 \frac{d}{d x} [\ln (\frac{1}{x})]$

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 $g (x) = \frac{1}{x}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{1}{x}$ .

$3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{1}{x}])$

Step 2.2

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

$3 (\frac{1}{u} \frac{d}{d x} [\frac{1}{x}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{1}{x}$ .

$3 (\frac{1}{\frac{1}{x}} \frac{d}{d x} [\frac{1}{x}])$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{x}$ .

$3 (1 x \frac{d}{d x} [\frac{1}{x}])$

Step 4

Simplify the expression.

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Step 4.1

Multiply $x$ by $1$ .

$3 (x \frac{d}{d x} [\frac{1}{x}])$

Step 4.2

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

$3 x \frac{d}{d x} [x^{- 1}]$

Step 5

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

$3 x (- x^{- 2})$

Step 6

Multiply $- 1$ by $3$ .

$- 3 x \cdot x^{- 2}$

Step 7

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

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Step 7.1

Move $x^{- 2}$ .

$- 3 (x^{- 2} x)$

Step 7.2

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

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Step 7.2.1

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

$- 3 (x^{- 2} x^{1})$

Step 7.2.2

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

$- 3 x^{- 2 + 1}$

Step 7.3

Add $- 2$ and $1$ .

$- 3 x^{- 1}$

Step 8

Simplify.

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Step 8.1

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

$- 3 \frac{1}{x}$

Step 8.2

Combine terms.

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Step 8.2.1

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

$\frac{- 3}{x}$

Step 8.2.2

Move the negative in front of the fraction.

$- \frac{3}{x}$