Calculus Examples

$\frac{x^{2} - 6 x + 2}{2 x}$

Step 1

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x^{2} - 6 x + 2}{2 x}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [\frac{x^{2} - 6 x + 2}{x}]$ .

$\frac{1}{2} \frac{d}{d x} [\frac{x^{2} - 6 x + 2}{x}]$

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 $f (x) = x^{2} - 6 x + 2$ and $g (x) = x$ .

$\frac{1}{2} \cdot \frac{x \frac{d}{d x} [x^{2} - 6 x + 2] - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{2} - 6 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [2]$ .

$\frac{1}{2} \cdot \frac{x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [2]) - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$\frac{1}{2} \cdot \frac{x (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [2]) - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6 x$ with respect to $x$ is $- 6 \frac{d}{d x} [x]$ .

$\frac{1}{2} \cdot \frac{x (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [2]) - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4

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

$\frac{1}{2} \cdot \frac{x (2 x - 6 \cdot 1 + \frac{d}{d x} [2]) - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.5

Multiply $- 6$ by $1$ .

$\frac{1}{2} \cdot \frac{x (2 x - 6 + \frac{d}{d x} [2]) - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$\frac{1}{2} \cdot \frac{x (2 x - 6 + 0) - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.7

Add $2 x - 6$ and $0$ .

$\frac{1}{2} \cdot \frac{x (2 x - 6) - (x^{2} - 6 x + 2) \frac{d}{d x} [x]}{x^{2}}$

Step 3.8

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

$\frac{1}{2} \cdot \frac{x (2 x - 6) - (x^{2} - 6 x + 2) \cdot 1}{x^{2}}$

Step 3.9

Combine fractions.

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Step 3.9.1

Multiply $- 1$ by $1$ .

$\frac{1}{2} \cdot \frac{x (2 x - 6) - (x^{2} - 6 x + 2)}{x^{2}}$

Step 3.9.2

Multiply $\frac{1}{2}$ by $\frac{x (2 x - 6) - (x^{2} - 6 x + 2)}{x^{2}}$ .

$\frac{x (2 x - 6) - (x^{2} - 6 x + 2)}{2 x^{2}}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 6 - (x^{2} - 6 x + 2)}{2 x^{2}}$

Step 4.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 2}{2 x^{2}}$

Step 4.3

Simplify the numerator.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 2}{2 x^{2}}$

Step 4.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 2}{2 x^{2}}$

Step 4.3.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 2}{2 x^{2}}$

Step 4.3.1.3

Move $- 6$ to the left of $x$ .

$\frac{2 x^{2} - 6 \cdot x - x^{2} - (- 6 x) - 1 \cdot 2}{2 x^{2}}$

Step 4.3.1.4

Multiply $- 6$ by $- 1$ .

$\frac{2 x^{2} - 6 x - x^{2} + 6 x - 1 \cdot 2}{2 x^{2}}$

Step 4.3.1.5

Multiply $- 1$ by $2$ .

$\frac{2 x^{2} - 6 x - x^{2} + 6 x - 2}{2 x^{2}}$

Step 4.3.2

Combine the opposite terms in $2 x^{2} - 6 x - x^{2} + 6 x - 2$ .

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Step 4.3.2.1

Add $- 6 x$ and $6 x$ .

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

Step 4.3.2.2

Add $2 x^{2} - x^{2}$ and $0$ .

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

Step 4.3.3

Subtract $x^{2}$ from $2 x^{2}$ .

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