Calculus Examples

$- \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}}$ .

$- \frac{d}{d x} [\frac{x^{2} + 36}{{(x^{2} - 36)}^{2}}] + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

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 $f (x) = x^{2} + 36$ and $g (x) = {(x^{2} - 36)}^{2}$ .

$- \frac{{(x^{2} - 36)}^{2} \frac{d}{d x} [x^{2} + 36] - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{({(x^{2} - 36)}^{2})}^{2}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 3

Differentiate.

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Step 3.1

Multiply the exponents in ${({(x^{2} - 36)}^{2})}^{2}$ .

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Step 3.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- \frac{{(x^{2} - 36)}^{2} \frac{d}{d x} [x^{2} + 36] - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{(x^{2} - 36)}^{2 \cdot 2}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 3.1.2

Multiply $2$ by $2$ .

$- \frac{{(x^{2} - 36)}^{2} \frac{d}{d x} [x^{2} + 36] - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 3.2

By the Sum Rule, the derivative of $x^{2} + 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]$ .

$- \frac{{(x^{2} - 36)}^{2} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [36]) - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 3.3

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

$- \frac{{(x^{2} - 36)}^{2} (2 x + \frac{d}{d x} [36]) - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 3.4

Since $36$ is constant with respect to $x$ , the derivative of $36$ with respect to $x$ is $0$ .

$- \frac{{(x^{2} - 36)}^{2} (2 x + 0) - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $2 x$ and $0$ .

$- \frac{{(x^{2} - 36)}^{2} (2 x) - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 3.5.2

Move $2$ to the left of ${(x^{2} - 36)}^{2}$ .

$- \frac{2 \cdot {(x^{2} - 36)}^{2} x - (x^{2} + 36) \frac{d}{d x} [{(x^{2} - 36)}^{2}]}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 4

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 36$ .

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Step 4.1

To apply the Chain Rule, set $u$ as $x^{2} - 36$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - (x^{2} + 36) (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} - 36])}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 4.2

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

$- \frac{2 {(x^{2} - 36)}^{2} x - (x^{2} + 36) (2 u \frac{d}{d x} [x^{2} - 36])}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 4.3

Replace all occurrences of $u$ with $x^{2} - 36$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - (x^{2} + 36) (2 (x^{2} - 36) \frac{d}{d x} [x^{2} - 36])}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5

Differentiate.

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Step 5.1

Multiply $2$ by $- 1$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 2 (x^{2} + 36) ((x^{2} - 36) \frac{d}{d x} [x^{2} - 36])}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5.2

By the Sum Rule, the derivative of $x^{2} - 36$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 2 (x^{2} + 36) ((x^{2} - 36) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 36]))}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5.3

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

$- \frac{2 {(x^{2} - 36)}^{2} x - 2 (x^{2} + 36) ((x^{2} - 36) (2 x + \frac{d}{d x} [- 36]))}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5.4

Since $- 36$ is constant with respect to $x$ , the derivative of $- 36$ with respect to $x$ is $0$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 2 (x^{2} + 36) ((x^{2} - 36) (2 x + 0))}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5.5

Simplify the expression.

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Step 5.5.1

Add $2 x$ and $0$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 2 (x^{2} + 36) ((x^{2} - 36) (2 x))}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5.5.2

Move $2$ to the left of $x^{2} - 36$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 2 (x^{2} + 36) (2 \cdot (x^{2} - 36) x)}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5.5.3

Multiply $2$ by $- 2$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 4 (x^{2} + 36) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \frac{d}{d x} [- 1]$

Step 5.6

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

$- \frac{2 {(x^{2} - 36)}^{2} x - 4 (x^{2} + 36) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}} + \frac{x^{2} + 36}{{(x^{2} - 36)}^{2}} \cdot 0$

Step 5.7

Simplify the expression.

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Step 5.7.1

Multiply $\frac{x^{2} + 36}{{(x^{2} - 36)}^{2}}$ by $0$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 4 (x^{2} + 36) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}} + 0$

Step 5.7.2

Add $- \frac{2 {(x^{2} - 36)}^{2} x - 4 (x^{2} + 36) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}}$ and $0$ .

$- \frac{2 {(x^{2} - 36)}^{2} x - 4 (x^{2} + 36) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$- \frac{2 {(x^{2} - 36)}^{2} x + (- 4 x^{2} - 4 \cdot 36) ((x^{2} - 36) x)}{{(x^{2} - 36)}^{4}}$

Step 6.2

Apply the distributive property.

$- \frac{2 {(x^{2} - 36)}^{2} x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Rewrite ${(x^{2} - 36)}^{2}$ as $(x^{2} - 36) (x^{2} - 36)$ .

$- \frac{2 ((x^{2} - 36) (x^{2} - 36)) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.2

Expand $(x^{2} - 36) (x^{2} - 36)$ using the FOIL Method.

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Step 6.3.1.2.1

Apply the distributive property.

$- \frac{2 (x^{2} (x^{2} - 36) - 36 (x^{2} - 36)) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.2.2

Apply the distributive property.

$- \frac{2 (x^{2} x^{2} + x^{2} \cdot - 36 - 36 (x^{2} - 36)) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.2.3

Apply the distributive property.

$- \frac{2 (x^{2} x^{2} + x^{2} \cdot - 36 - 36 x^{2} - 36 \cdot - 36) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.3

Simplify and combine like terms.

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Step 6.3.1.3.1

Simplify each term.

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Step 6.3.1.3.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 6.3.1.3.1.1.1

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

$- \frac{2 (x^{2 + 2} + x^{2} \cdot - 36 - 36 x^{2} - 36 \cdot - 36) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.3.1.1.2

Add $2$ and $2$ .

$- \frac{2 (x^{4} + x^{2} \cdot - 36 - 36 x^{2} - 36 \cdot - 36) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.3.1.2

Move $- 36$ to the left of $x^{2}$ .

$- \frac{2 (x^{4} - 36 \cdot x^{2} - 36 x^{2} - 36 \cdot - 36) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.3.1.3

Multiply $- 36$ by $- 36$ .

$- \frac{2 (x^{4} - 36 x^{2} - 36 x^{2} + 1296) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.3.2

Subtract $36 x^{2}$ from $- 36 x^{2}$ .

$- \frac{2 (x^{4} - 72 x^{2} + 1296) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.4

Apply the distributive property.

$- \frac{(2 x^{4} + 2 (- 72 x^{2}) + 2 \cdot 1296) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.5

Simplify.

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Step 6.3.1.5.1

Multiply $- 72$ by $2$ .

$- \frac{(2 x^{4} - 144 x^{2} + 2 \cdot 1296) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.5.2

Multiply $2$ by $1296$ .

$- \frac{(2 x^{4} - 144 x^{2} + 2592) x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.6

Apply the distributive property.

$- \frac{2 x^{4} x - 144 x^{2} x + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7

Simplify.

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Step 6.3.1.7.1

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 6.3.1.7.1.1

Move $x$ .

$- \frac{2 (x \cdot x^{4}) - 144 x^{2} x + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7.1.2

Multiply $x$ by $x^{4}$ .

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Step 6.3.1.7.1.2.1

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

$- \frac{2 (x^{1} x^{4}) - 144 x^{2} x + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7.1.2.2

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

$- \frac{2 x^{1 + 4} - 144 x^{2} x + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7.1.3

Add $1$ and $4$ .

$- \frac{2 x^{5} - 144 x^{2} x + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.3.1.7.2.1

Move $x$ .

$- \frac{2 x^{5} - 144 (x \cdot x^{2}) + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7.2.2

Multiply $x$ by $x^{2}$ .

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Step 6.3.1.7.2.2.1

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

$- \frac{2 x^{5} - 144 (x^{1} x^{2}) + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7.2.2.2

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

$- \frac{2 x^{5} - 144 x^{1 + 2} + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.7.2.3

Add $1$ and $2$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x + (- 4 x^{2} - 4 \cdot 36) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.8

Multiply $- 4$ by $36$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x + (- 4 x^{2} - 144) (x^{2} x - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.9

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.3.1.9.1

Multiply $x^{2}$ by $x$ .

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Step 6.3.1.9.1.1

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

$- \frac{2 x^{5} - 144 x^{3} + 2592 x + (- 4 x^{2} - 144) (x^{2} x^{1} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.9.1.2

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

$- \frac{2 x^{5} - 144 x^{3} + 2592 x + (- 4 x^{2} - 144) (x^{2 + 1} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.9.2

Add $2$ and $1$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x + (- 4 x^{2} - 144) (x^{3} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.10

Expand $(- 4 x^{2} - 144) (x^{3} - 36 x)$ using the FOIL Method.

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Step 6.3.1.10.1

Apply the distributive property.

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{2} (x^{3} - 36 x) - 144 (x^{3} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.10.2

Apply the distributive property.

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{2} x^{3} - 4 x^{2} (- 36 x) - 144 (x^{3} - 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.10.3

Apply the distributive property.

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{2} x^{3} - 4 x^{2} (- 36 x) - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11

Simplify and combine like terms.

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Step 6.3.1.11.1

Simplify each term.

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Step 6.3.1.11.1.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 6.3.1.11.1.1.1

Move $x^{3}$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 (x^{3} x^{2}) - 4 x^{2} (- 36 x) - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.1.2

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

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{3 + 2} - 4 x^{2} (- 36 x) - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.1.3

Add $3$ and $2$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} - 4 x^{2} (- 36 x) - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.2

Rewrite using the commutative property of multiplication.

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} - 4 \cdot - 36 x^{2} x - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.3.1.11.1.3.1

Move $x$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} - 4 \cdot - 36 (x \cdot x^{2}) - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 6.3.1.11.1.3.2.1

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

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} - 4 \cdot - 36 (x^{1} x^{2}) - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.3.2.2

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

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} - 4 \cdot - 36 x^{1 + 2} - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.3.3

Add $1$ and $2$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} - 4 \cdot - 36 x^{3} - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.4

Multiply $- 4$ by $- 36$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} + 144 x^{3} - 144 x^{3} - 144 (- 36 x)}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.1.5

Multiply $- 36$ by $- 144$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} + 144 x^{3} - 144 x^{3} + 5184 x}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.2

Subtract $144 x^{3}$ from $144 x^{3}$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} + 0 + 5184 x}{{(x^{2} - 36)}^{4}}$

Step 6.3.1.11.3

Add $- 4 x^{5}$ and $0$ .

$- \frac{2 x^{5} - 144 x^{3} + 2592 x - 4 x^{5} + 5184 x}{{(x^{2} - 36)}^{4}}$

Step 6.3.2

Subtract $4 x^{5}$ from $2 x^{5}$ .

$- \frac{- 2 x^{5} - 144 x^{3} + 2592 x + 5184 x}{{(x^{2} - 36)}^{4}}$

Step 6.3.3

Add $2592 x$ and $5184 x$ .

$- \frac{- 2 x^{5} - 144 x^{3} + 7776 x}{{(x^{2} - 36)}^{4}}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Factor $2 x$ out of $- 2 x^{5} - 144 x^{3} + 7776 x$ .

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Step 6.4.1.1

Factor $2 x$ out of $- 2 x^{5}$ .

$- \frac{2 x (- x^{4}) - 144 x^{3} + 7776 x}{{(x^{2} - 36)}^{4}}$

Step 6.4.1.2

Factor $2 x$ out of $- 144 x^{3}$ .

$- \frac{2 x (- x^{4}) + 2 x (- 72 x^{2}) + 7776 x}{{(x^{2} - 36)}^{4}}$

Step 6.4.1.3

Factor $2 x$ out of $7776 x$ .

$- \frac{2 x (- x^{4}) + 2 x (- 72 x^{2}) + 2 x (3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.1.4

Factor $2 x$ out of $2 x (- x^{4}) + 2 x (- 72 x^{2})$ .

$- \frac{2 x (- x^{4} - 72 x^{2}) + 2 x (3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.1.5

Factor $2 x$ out of $2 x (- x^{4} - 72 x^{2}) + 2 x (3888)$ .

$- \frac{2 x (- x^{4} - 72 x^{2} + 3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$- \frac{2 x (- {(x^{2})}^{2} - 72 x^{2} + 3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$- \frac{2 x (- u^{2} - 72 u + 3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.4

Factor by grouping.

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Step 6.4.4.1

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 $a \cdot c = - 1 \cdot 3888 = - 3888$ and whose sum is $b = - 72$ .

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Step 6.4.4.1.1

Factor $- 72$ out of $- 72 u$ .

$- \frac{2 x (- u^{2} - 72 (u) + 3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.4.1.2

Rewrite $- 72$ as $36$ plus $- 108$

$- \frac{2 x (- u^{2} + (36 - 108) u + 3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.4.1.3

Apply the distributive property.

$- \frac{2 x (- u^{2} + 36 u - 108 u + 3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.4.2

Factor out the greatest common factor from each group.

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Step 6.4.4.2.1

Group the first two terms and the last two terms.

$- \frac{2 x ((- u^{2} + 36 u) - 108 u + 3888)}{{(x^{2} - 36)}^{4}}$

Step 6.4.4.2.2

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

$- \frac{2 x (u (- u + 36) + 108 (- u + 36))}{{(x^{2} - 36)}^{4}}$

Step 6.4.4.3

Factor the polynomial by factoring out the greatest common factor, $- u + 36$ .

$- \frac{2 x ((- u + 36) (u + 108))}{{(x^{2} - 36)}^{4}}$

Step 6.4.5

Replace all occurrences of $u$ with $x^{2}$ .

$- \frac{2 x ((- x^{2} + 36) (x^{2} + 108))}{{(x^{2} - 36)}^{4}}$

Step 6.4.6

Rewrite $36$ as $6^{2}$ .

$- \frac{2 x ((- x^{2} + 6^{2}) (x^{2} + 108))}{{(x^{2} - 36)}^{4}}$

Step 6.4.7

Reorder $- x^{2}$ and $6^{2}$ .

$- \frac{2 x ((6^{2} - x^{2}) (x^{2} + 108))}{{(x^{2} - 36)}^{4}}$

Step 6.4.8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

$- \frac{2 x (6 + x) (6 - x) (x^{2} + 108)}{{(x^{2} - 36)}^{4}}$

Step 6.5

Simplify the denominator.

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Step 6.5.1

Rewrite $36$ as $6^{2}$ .

$- \frac{2 x (6 + x) (6 - x) (x^{2} + 108)}{{(x^{2} - 6^{2})}^{4}}$

Step 6.5.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

$- \frac{2 x (6 + x) (6 - x) (x^{2} + 108)}{{((x + 6) (x - 6))}^{4}}$

Step 6.5.3

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

$- \frac{2 x (6 + x) (6 - x) (x^{2} + 108)}{{(x + 6)}^{4} {(x - 6)}^{4}}$

Step 6.6

Cancel the common factor of $6 + x$ and ${(x + 6)}^{4}$ .

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Step 6.6.1

Reorder terms.

$- \frac{2 x (x + 6) (6 - x) (x^{2} + 108)}{{(x + 6)}^{4} {(x - 6)}^{4}}$

Step 6.6.2

Factor $x + 6$ out of $2 x (x + 6) (6 - x) (x^{2} + 108)$ .

$- \frac{(x + 6) ((2 x (6 - x)) (x^{2} + 108))}{{(x + 6)}^{4} {(x - 6)}^{4}}$

Step 6.6.3

Cancel the common factors.

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Step 6.6.3.1

Factor $x + 6$ out of ${(x + 6)}^{4} {(x - 6)}^{4}$ .

$- \frac{(x + 6) ((2 x (6 - x)) (x^{2} + 108))}{(x + 6) ({(x + 6)}^{3} {(x - 6)}^{4})}$

Step 6.6.3.2

Cancel the common factor.

$- \frac{(x + 6) ((2 x (6 - x)) (x^{2} + 108))}{(x + 6) ({(x + 6)}^{3} {(x - 6)}^{4})}$

Step 6.6.3.3

Rewrite the expression.

$- \frac{(2 x (6 - x)) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 6.7

Cancel the common factor of $6 - x$ and ${(x - 6)}^{4}$ .

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Step 6.7.1

Rewrite $6$ as $- 1 (- 6)$ .

$- \frac{2 x (- 1 (- 6) - x) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 6.7.2

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

$- \frac{2 x (- 1 (- 6) - (x)) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 6.7.3

Factor $- 1$ out of $- 1 (- 6) - (x)$ .

$- \frac{2 x (- 1 (- 6 + x)) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 6.7.4

Reorder terms.

$- \frac{2 x (- 1 (x - 6)) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 6.7.5

Factor $x - 6$ out of $2 x (- 1 (x - 6)) (x^{2} + 108)$ .

$- \frac{(x - 6) (2 x \cdot (- 1) (x^{2} + 108))}{{(x + 6)}^{3} {(x - 6)}^{4}}$

Step 6.7.6

Cancel the common factors.

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Step 6.7.6.1

Factor $x - 6$ out of ${(x + 6)}^{3} {(x - 6)}^{4}$ .

$- \frac{(x - 6) (2 x \cdot (- 1) (x^{2} + 108))}{(x - 6) ({(x + 6)}^{3} {(x - 6)}^{3})}$

Step 6.7.6.2

Cancel the common factor.

$- \frac{(x - 6) (2 x \cdot (- 1) (x^{2} + 108))}{(x - 6) ({(x + 6)}^{3} {(x - 6)}^{3})}$

Step 6.7.6.3

Rewrite the expression.

$- \frac{2 x \cdot (- 1) (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$

Step 6.8

Multiply $- 1$ by $2$ .

$- \frac{- 2 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$

Step 6.9

Move the negative in front of the fraction.

$- - \frac{2 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$

Step 6.10

Multiply $- - \frac{2 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$ .

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Step 6.10.1

Multiply $- 1$ by $- 1$ .

$1 \frac{2 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$

Step 6.10.2

Multiply $\frac{2 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$ by $1$ .

$\frac{2 x (x^{2} + 108)}{{(x + 6)}^{3} {(x - 6)}^{3}}$