Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \ln (x)$ and $g (x) = x$ .

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

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor of $x$ .

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Step 3.3.2.1

Cancel the common factor.

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

Step 3.3.2.2

Rewrite the expression.

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

Step 3.3.3

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

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

Step 3.3.4

Multiply $- 1$ by $1$ .

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

Step 4

Reform the equation by setting the left side equal to the right side.

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

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

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