Calculus Examples

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

Step 1

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

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

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x \ln (x)$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x \ln (x)$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $x \ln (x)$ .

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

Step 3

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = \ln (x)$ .

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

Step 4

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

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

Step 5

Differentiate using the Power Rule.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

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

$- {(x \ln (x))}^{- 2} (1 + \ln (x) \cdot 1)$

Step 5.4

Multiply $\ln (x)$ by $1$ .

$- {(x \ln (x))}^{- 2} (1 + \ln (x))$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Apply the product rule to $x \ln (x)$ .

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Combine terms.

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Step 7.3.1

Multiply $- 1$ by $1$ .

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

Step 7.3.2

Combine $\ln (x)$ and $\frac{1}{x^{2} \ln^{2} (x)}$ .

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

Step 7.3.3

Cancel the common factor of $\ln (x)$ and $\ln^{2} (x)$ .

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Step 7.3.3.1

Multiply by $1$ .

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

Step 7.3.3.2

Cancel the common factors.

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Step 7.3.3.2.1

Factor $\ln (x)$ out of $x^{2} \ln^{2} (x)$ .

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

Step 7.3.3.2.2

Cancel the common factor.

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

Step 7.3.3.2.3

Rewrite the expression.

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

Step 7.4

Reorder terms.

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