Calculus Examples

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

Step 1

Factor $x$ out of $x^{2} - 3 x$ .

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Step 1.1

Factor $x$ out of $x^{2}$ .

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

Step 1.2

Factor $x$ out of $- 3 x$ .

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

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot - 3$ .

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

Step 2

Cancel the common factors.

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Step 2.1

Factor $x$ out of $x^{2}$ .

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

Step 2.2

Cancel the common factor.

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

Step 2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $1$ and $0$ .

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

Step 4.4.2

Multiply $x$ by $1$ .

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

Step 4.5

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

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

Step 4.6

Multiply $- 1$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

Subtract $x$ from $x$ .

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

Step 5.2.2

Subtract $- 3$ from $0$ .

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

Step 5.2.3

Multiply $- 1$ by $- 3$ .

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