Calculus Examples

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

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

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

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

Step 5.2

Cancel the common factor of $y$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 6

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

$\frac{1}{x} \cdot 1$

Step 7

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

$\frac{1}{x}$