Calculus Examples

$\frac{x^{2} - 6 x + 12}{x - 4}$

Step 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 $f (x) = x^{2} - 6 x + 12$ and $g (x) = x - 4$ .

$\frac{(x - 4) \frac{d}{d x} [x^{2} - 6 x + 12] - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - 6 x + 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [12]$ .

$\frac{(x - 4) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [12]) - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2.2

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

$\frac{(x - 4) (2 x + \frac{d}{d x} [- 6 x] + \frac{d}{d x} [12]) - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2.3

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

$\frac{(x - 4) (2 x - 6 \frac{d}{d x} [x] + \frac{d}{d x} [12]) - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2.4

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

$\frac{(x - 4) (2 x - 6 \cdot 1 + \frac{d}{d x} [12]) - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2.5

Multiply $- 6$ by $1$ .

$\frac{(x - 4) (2 x - 6 + \frac{d}{d x} [12]) - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2.6

Since $12$ is constant with respect to $x$ , the derivative of $12$ with respect to $x$ is $0$ .

$\frac{(x - 4) (2 x - 6 + 0) - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2.7

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

$\frac{(x - 4) (2 x - 6) - (x^{2} - 6 x + 12) \frac{d}{d x} [x - 4]}{{(x - 4)}^{2}}$

Step 2.8

By the Sum Rule, the derivative of $x - 4$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 4]$ .

$\frac{(x - 4) (2 x - 6) - (x^{2} - 6 x + 12) (\frac{d}{d x} [x] + \frac{d}{d x} [- 4])}{{(x - 4)}^{2}}$

Step 2.9

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

$\frac{(x - 4) (2 x - 6) - (x^{2} - 6 x + 12) (1 + \frac{d}{d x} [- 4])}{{(x - 4)}^{2}}$

Step 2.10

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$\frac{(x - 4) (2 x - 6) - (x^{2} - 6 x + 12) (1 + 0)}{{(x - 4)}^{2}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

$\frac{(x - 4) (2 x - 6) - (x^{2} - 6 x + 12) \cdot 1}{{(x - 4)}^{2}}$

Step 2.11.2

Multiply $- 1$ by $1$ .

$\frac{(x - 4) (2 x - 6) - (x^{2} - 6 x + 12)}{{(x - 4)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$\frac{(x - 4) (2 x - 6) - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(x - 4) (2 x - 6)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

$\frac{x (2 x - 6) - 4 (2 x - 6) - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.1.2

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 6 - 4 (2 x - 6) - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.1.3

Apply the distributive property.

$\frac{x (2 x) + x \cdot - 6 - 4 (2 x) - 4 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x \cdot x + x \cdot - 6 - 4 (2 x) - 4 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x$ .

$\frac{2 (x \cdot x) + x \cdot - 6 - 4 (2 x) - 4 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

$\frac{2 x^{2} + x \cdot - 6 - 4 (2 x) - 4 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.2.1.3

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

$\frac{2 x^{2} - 6 \cdot x - 4 (2 x) - 4 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.2.1.4

Multiply $2$ by $- 4$ .

$\frac{2 x^{2} - 6 x - 8 x - 4 \cdot - 6 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.2.1.5

Multiply $- 4$ by $- 6$ .

$\frac{2 x^{2} - 6 x - 8 x + 24 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.2.2

Subtract $8 x$ from $- 6 x$ .

$\frac{2 x^{2} - 14 x + 24 - x^{2} - (- 6 x) - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.3

Multiply $- 6$ by $- 1$ .

$\frac{2 x^{2} - 14 x + 24 - x^{2} + 6 x - 1 \cdot 12}{{(x - 4)}^{2}}$

Step 3.2.1.4

Multiply $- 1$ by $12$ .

$\frac{2 x^{2} - 14 x + 24 - x^{2} + 6 x - 12}{{(x - 4)}^{2}}$

Step 3.2.2

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

$\frac{x^{2} - 14 x + 24 + 6 x - 12}{{(x - 4)}^{2}}$

Step 3.2.3

Add $- 14 x$ and $6 x$ .

$\frac{x^{2} - 8 x + 24 - 12}{{(x - 4)}^{2}}$

Step 3.2.4

Subtract $12$ from $24$ .

$\frac{x^{2} - 8 x + 12}{{(x - 4)}^{2}}$

Step 3.3

Factor $x^{2} - 8 x + 12$ using the AC method.

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Step 3.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 3.3.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 2)}{{(x - 4)}^{2}}$