Calculus Examples

$\log (\frac{1}{x})$

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\log (u)$ with respect to $u$ is $\frac{1}{u \ln (10)}$ .

$\frac{1}{u \ln (10)} \frac{d}{d x} [\frac{1}{x}]$

Step 1.3

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

$\frac{1}{\frac{1}{x} \ln (10)} \frac{d}{d x} [\frac{1}{x}]$

Step 2

Combine $\frac{1}{x}$ and $\ln (10)$ .

$\frac{1}{\frac{\ln (10)}{x}} \frac{d}{d x} [\frac{1}{x}]$

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{\ln (10)}{x}$ .

$1 \frac{x}{\ln (10)} \frac{d}{d x} [\frac{1}{x}]$

Step 4

Simplify the expression.

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Step 4.1

Multiply $\frac{x}{\ln (10)}$ by $1$ .

$\frac{x}{\ln (10)} \frac{d}{d x} [\frac{1}{x}]$

Step 4.2

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

$\frac{x}{\ln (10)} \frac{d}{d x} [x^{- 1}]$

Step 5

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

$\frac{x}{\ln (10)} (- x^{- 2})$

Step 6

Combine $\frac{x}{\ln (10)}$ and $x^{- 2}$ .

$- \frac{x \cdot x^{- 2}}{\ln (10)}$

Step 7

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

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Step 7.1

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

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Step 7.1.1

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

$- \frac{x^{1} x^{- 2}}{\ln (10)}$

Step 7.1.2

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

$- \frac{x^{1 - 2}}{\ln (10)}$

Step 7.2

Subtract $2$ from $1$ .

$- \frac{x^{- 1}}{\ln (10)}$

Step 8

Move $x^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{\ln (10) x}$

Step 9

Reorder factors in $- \frac{1}{\ln (10) x}$ .

$- \frac{1}{x \ln (10)}$