Calculus Examples

$y = \ln (\frac{3}{x})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \ln (x)$ and $g (x) = \frac{3}{x}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $\frac{3}{x}$ .

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

Step 1.2

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

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

Step 1.3

Replace all occurrences of $u$ with $\frac{3}{x}$ .

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

Step 2

Multiply by the reciprocal of the fraction to divide by $\frac{3}{x}$ .

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

Step 3

Multiply $\frac{x}{3}$ by $1$ .

$\frac{x}{3} \frac{d}{d x} [\frac{3}{x}]$

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $3$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Divide $x$ by $1$ .

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

$- x^{1 - 2}$

Step 9

Subtract $2$ from $1$ .

$- x^{- 1}$

Step 10

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

$- \frac{1}{x}$