Calculus Examples

$\frac{1 - 3 x}{3 x}$

Step 1

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

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

Step 2

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 $f (x) = 1 - 3 x$ and $g (x) = x$ .

$\frac{1}{3} \cdot \frac{x \frac{d}{d x} [1 - 3 x] - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $1 - 3 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]$ .

$\frac{1}{3} \cdot \frac{x (\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]) - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{1}{3} \cdot \frac{x (0 + \frac{d}{d x} [- 3 x]) - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{1}{3} \cdot \frac{x \frac{d}{d x} [- 3 x] - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4

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

$\frac{1}{3} \cdot \frac{x (- 3 \frac{d}{d x} [x]) - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.5

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

$\frac{1}{3} \cdot \frac{x (- 3 \cdot 1) - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6

Simplify the expression.

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Step 3.6.1

Multiply $- 3$ by $1$ .

$\frac{1}{3} \cdot \frac{x \cdot - 3 - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6.2

Move $- 3$ to the left of $x$ .

$\frac{1}{3} \cdot \frac{- 3 \cdot x - (1 - 3 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.7

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

$\frac{1}{3} \cdot \frac{- 3 x - (1 - 3 x) \cdot 1}{x^{2}}$

Step 3.8

Combine fractions.

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Step 3.8.1

Multiply $- 1$ by $1$ .

$\frac{1}{3} \cdot \frac{- 3 x - (1 - 3 x)}{x^{2}}$

Step 3.8.2

Multiply $\frac{1}{3}$ by $\frac{- 3 x - (1 - 3 x)}{x^{2}}$ .

$\frac{- 3 x - (1 - 3 x)}{3 x^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{- 3 x - 1 \cdot 1 - (- 3 x)}{3 x^{2}}$

Step 4.2

Simplify the numerator.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $- 1$ by $1$ .

$\frac{- 3 x - 1 - (- 3 x)}{3 x^{2}}$

Step 4.2.1.2

Multiply $- 3$ by $- 1$ .

$\frac{- 3 x - 1 + 3 x}{3 x^{2}}$

Step 4.2.2

Combine the opposite terms in $- 3 x - 1 + 3 x$ .

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Step 4.2.2.1

Add $- 3 x$ and $3 x$ .

$\frac{0 - 1}{3 x^{2}}$

Step 4.2.2.2

Subtract $1$ from $0$ .

$\frac{- 1}{3 x^{2}}$

Step 4.3

Move the negative in front of the fraction.

$- \frac{1}{3 x^{2}}$