Calculus Examples

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

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Add $1$ and $0$ .

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

Step 2.4.2

Multiply $x^{2} + 4 x - 12$ by $1$ .

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

Step 2.5

By the Sum Rule, the derivative of $x^{2} + 4 x - 12$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [4 x] + \frac{d}{d x} [- 12]$ .

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

Step 2.6

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

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

Step 2.7

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

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

Step 2.8

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

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

Step 2.9

Multiply $4$ by $1$ .

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

Step 2.10

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

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

Step 2.11

Add $2 x + 4$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

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

Multiply $- 1$ by $- 2$ .

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

Step 3.2.1.2

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

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Step 3.2.1.2.1

Apply the distributive property.

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

Step 3.2.1.2.2

Apply the distributive property.

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

Step 3.2.1.2.3

Apply the distributive property.

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

Step 3.2.1.3

Simplify and combine like terms.

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Step 3.2.1.3.1

Simplify each term.

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Step 3.2.1.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.3.1.2

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

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Step 3.2.1.3.1.2.1

Move $x$ .

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

Step 3.2.1.3.1.2.2

Multiply $x$ by $x$ .

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

Step 3.2.1.3.1.3

Multiply $- 1$ by $2$ .

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

Step 3.2.1.3.1.4

Multiply $4$ by $- 1$ .

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

Step 3.2.1.3.1.5

Multiply $2$ by $2$ .

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

Step 3.2.1.3.1.6

Multiply $2$ by $4$ .

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

Step 3.2.1.3.2

Add $- 4 x$ and $4 x$ .

$\frac{x^{2} + 4 x - 12 - 2 x^{2} + 0 + 8}{{(x^{2} + 4 x - 12)}^{2}}$

Step 3.2.1.3.3

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

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

Step 3.2.2

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

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

Step 3.2.3

Add $- 12$ and $8$ .

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

Step 3.3

Factor by grouping.

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Step 3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 4 = 4$ and whose sum is $b = 4$ .

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Step 3.3.1.1

Factor $4$ out of $4 x$ .

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

Step 3.3.1.2

Rewrite $4$ as $2$ plus $2$

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

Step 3.3.1.3

Apply the distributive property.

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

Step 3.3.2

Factor out the greatest common factor from each group.

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Step 3.3.2.1

Group the first two terms and the last two terms.

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

Step 3.3.2.2

Factor out the greatest common factor (GCF) from each group.

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

Step 3.3.3

Factor the polynomial by factoring out the greatest common factor, $- x + 2$ .

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

Step 3.4

Simplify the denominator.

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Step 3.4.1

Factor $x^{2} + 4 x - 12$ using the AC method.

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Step 3.4.1.1

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 $4$ .

$- 2, 6$

Step 3.4.1.2

Write the factored form using these integers.

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

Step 3.4.2

Apply the product rule to $(x - 2) (x + 6)$ .

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

Step 3.5

Simplify the numerator.

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Step 3.5.1

Factor $- 1$ out of $- x$ .

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

Step 3.5.2

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 3.5.3

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 3.5.4

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 3.5.5

Raise $x - 2$ to the power of $1$ .

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

Step 3.5.6

Raise $x - 2$ to the power of $1$ .

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

Step 3.5.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5.8

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of ${(x - 2)}^{2}$ .

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Step 3.6.1

Cancel the common factor.

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

Step 3.6.2

Rewrite the expression.

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

Step 3.7

Move the negative in front of the fraction.

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