Calculus Examples

$f (x) = 2 + 3 \ln (\frac{1}{x})$

Step 1

Differentiate.

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Step 1.1

By the Sum Rule, the derivative of $2 + 3 \ln (\frac{1}{x})$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [3 \ln (\frac{1}{x})]$ .

$\frac{d}{d x} [2] + \frac{d}{d x} [3 \ln (\frac{1}{x})]$

Step 1.2

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [3 \ln (\frac{1}{x})]$

Step 2

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

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Step 2.1

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

$0 + 3 \frac{d}{d x} [\ln (\frac{1}{x})]$

Step 2.2

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 2.2.1

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

$0 + 3 (\frac{d}{d u} [\ln (u)] \frac{d}{d x} [\frac{1}{x}])$

Step 2.2.2

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

$0 + 3 (\frac{1}{u} \frac{d}{d x} [\frac{1}{x}])$

Step 2.2.3

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

$0 + 3 (\frac{1}{\frac{1}{x}} \frac{d}{d x} [\frac{1}{x}])$

Step 2.3

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

$0 + 3 (\frac{1}{\frac{1}{x}} \frac{d}{d x} [x^{- 1}])$

Step 2.4

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

$0 + 3 (\frac{1}{\frac{1}{x}} (- x^{- 2}))$

Step 2.5

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

$0 + 3 (1 x (- x^{- 2}))$

Step 2.6

Multiply $x$ by $1$ .

$0 + 3 (x (- x^{- 2}))$

Step 2.7

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

$0 + 3 (- (x^{1} x^{- 2}))$

Step 2.8

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

$0 + 3 (- x^{1 - 2})$

Step 2.9

Subtract $2$ from $1$ .

$0 + 3 (- x^{- 1})$

Step 2.10

Multiply $- 1$ by $3$ .

$0 - 3 x^{- 1}$

Step 3

Simplify.

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Step 3.1

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

$0 - 3 \frac{1}{x}$

Step 3.2

Combine terms.

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Step 3.2.1

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

$0 + \frac{- 3}{x}$

Step 3.2.2

Move the negative in front of the fraction.

$0 - \frac{3}{x}$

Step 3.2.3

Subtract $\frac{3}{x}$ from $0$ .

$- \frac{3}{x}$