Calculus Examples

$z = \frac{4 - 3 x}{3 x^{2} + x}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d z} (z) = \frac{d}{d z} (\frac{4 - 3 x}{3 x^{2} + x})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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Step 3.1.1

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

$\frac{d}{d z} [\frac{4 - 3 x}{x (3 x) + x}]$

Step 3.1.2

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

$\frac{d}{d z} [\frac{4 - 3 x}{x (3 x) + x^{1}}]$

Step 3.1.3

Factor $x$ out of $x^{1}$ .

$\frac{d}{d z} [\frac{4 - 3 x}{x (3 x) + x \cdot 1}]$

Step 3.1.4

Factor $x$ out of $x (3 x) + x \cdot 1$ .

$\frac{d}{d z} [\frac{4 - 3 x}{x (3 x + 1)}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d z} [\frac{f (z)}{g (z)}]$ is $\frac{g (z) \frac{d}{d z} [f (z)] - f (z) \frac{d}{d z} [g (z)]}{{g (z)}^{2}}$ where $f (z) = 4 - 3 x$ and $g (z) = x (3 x + 1)$ .

$\frac{x (3 x + 1) \frac{d}{d z} [4 - 3 x] - (4 - 3 x) \frac{d}{d z} [x (3 x + 1)]}{{(x (3 x + 1))}^{2}}$

Step 3.3

Differentiate.

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Step 3.3.1

By the Sum Rule, the derivative of $4 - 3 x$ with respect to $z$ is $\frac{d}{d z} [4] + \frac{d}{d z} [- 3 x]$ .

$\frac{x (3 x + 1) (\frac{d}{d z} [4] + \frac{d}{d z} [- 3 x]) - (4 - 3 x) \frac{d}{d z} [x (3 x + 1)]}{{(x (3 x + 1))}^{2}}$

Step 3.3.2

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

$\frac{x (3 x + 1) (0 + \frac{d}{d z} [- 3 x]) - (4 - 3 x) \frac{d}{d z} [x (3 x + 1)]}{{(x (3 x + 1))}^{2}}$

Step 3.3.3

Add $0$ and $\frac{d}{d z} [- 3 x]$ .

$\frac{x (3 x + 1) \frac{d}{d z} [- 3 x] - (4 - 3 x) \frac{d}{d z} [x (3 x + 1)]}{{(x (3 x + 1))}^{2}}$

Step 3.3.4

Since $- 3$ is constant with respect to $z$ , the derivative of $- 3 x$ with respect to $z$ is $- 3 \frac{d}{d z} [x]$ .

$\frac{x (3 x + 1) (- 3 \frac{d}{d z} [x]) - (4 - 3 x) \frac{d}{d z} [x (3 x + 1)]}{{(x (3 x + 1))}^{2}}$

Step 3.4

Rewrite $\frac{d}{d z} [x]$ as $x'$ .

$\frac{x (3 x + 1) (- 3 x') - (4 - 3 x) \frac{d}{d z} [x (3 x + 1)]}{{(x (3 x + 1))}^{2}}$

Step 3.5

Differentiate using the Product Rule which states that $\frac{d}{d z} [f (z) g (z)]$ is $f (z) \frac{d}{d z} [g (z)] + g (z) \frac{d}{d z} [f (z)]$ where $f (z) = x$ and $g (z) = 3 x + 1$ .

$\frac{x (3 x + 1) (- 3 x') - (4 - 3 x) (x \frac{d}{d z} [3 x + 1] + (3 x + 1) \frac{d}{d z} [x])}{{(x (3 x + 1))}^{2}}$

Step 3.6

Differentiate.

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Step 3.6.1

By the Sum Rule, the derivative of $3 x + 1$ with respect to $z$ is $\frac{d}{d z} [3 x] + \frac{d}{d z} [1]$ .

$\frac{x (3 x + 1) (- 3 x') - (4 - 3 x) (x (\frac{d}{d z} [3 x] + \frac{d}{d z} [1]) + (3 x + 1) \frac{d}{d z} [x])}{{(x (3 x + 1))}^{2}}$

Step 3.6.2

Since $3$ is constant with respect to $z$ , the derivative of $3 x$ with respect to $z$ is $3 \frac{d}{d z} [x]$ .

$\frac{x (3 x + 1) (- 3 x') - (4 - 3 x) (x (3 \frac{d}{d z} [x] + \frac{d}{d z} [1]) + (3 x + 1) \frac{d}{d z} [x])}{{(x (3 x + 1))}^{2}}$

Step 3.7

Rewrite $\frac{d}{d z} [x]$ as $x'$ .

$\frac{x (3 x + 1) (- 3 x') - (4 - 3 x) (x (3 x' + \frac{d}{d z} [1]) + (3 x + 1) \frac{d}{d z} [x])}{{(x (3 x + 1))}^{2}}$

Step 3.8

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

$\frac{x (3 x + 1) (- 3 x') - (4 - 3 x) (x (3 x' + 0) + (3 x + 1) \frac{d}{d z} [x])}{{(x (3 x + 1))}^{2}}$

Step 3.9

Add $3 x'$ and $0$ .

$\frac{x (3 x + 1) (- 3 x') - (4 - 3 x) (x (3 x') + (3 x + 1) \frac{d}{d z} [x])}{{(x (3 x + 1))}^{2}}$

Step 3.10

Rewrite $\frac{d}{d z} [x]$ as $x'$ .

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

Step 3.11

Simplify.

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Step 3.11.1

Apply the product rule to $x (3 x + 1)$ .

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

Step 3.11.2

Apply the distributive property.

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

Step 3.11.3

Apply the distributive property.

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

Step 3.11.4

Apply the distributive property.

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

Step 3.11.5

Apply the distributive property.

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

Step 3.11.6

Simplify the numerator.

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Step 3.11.6.1

Simplify each term.

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Step 3.11.6.1.1

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

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Step 3.11.6.1.1.1

Move $x$ .

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

Step 3.11.6.1.1.2

Multiply $x$ by $x$ .

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

Step 3.11.6.1.2

Move $3$ to the left of $x^{2}$ .

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

Step 3.11.6.1.3

Multiply $- 3$ by $3$ .

$\frac{- 9 x^{2} x' + x \cdot 1 (- 3 x') + (- 1 \cdot 4 - (- 3 x)) (x (3 x') + 3 x x' + 1 x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.4

Rewrite using the commutative property of multiplication.

$\frac{- 9 x^{2} x' - 3 x x' + (- 1 \cdot 4 - (- 3 x)) (x (3 x') + 3 x x' + 1 x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.5

Simplify each term.

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Step 3.11.6.1.5.1

Multiply $- 1$ by $4$ .

$\frac{- 9 x^{2} x' - 3 x x' + (- 4 - (- 3 x)) (x (3 x') + 3 x x' + 1 x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.5.2

Multiply $- 3$ by $- 1$ .

$\frac{- 9 x^{2} x' - 3 x x' + (- 4 + 3 x) (x (3 x') + 3 x x' + 1 x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.6

Simplify each term.

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Step 3.11.6.1.6.1

Rewrite using the commutative property of multiplication.

$\frac{- 9 x^{2} x' - 3 x x' + (- 4 + 3 x) (3 x x' + 3 x x' + 1 x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.6.2

Multiply $x'$ by $1$ .

$\frac{- 9 x^{2} x' - 3 x x' + (- 4 + 3 x) (3 x x' + 3 x x' + x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.7

Add $3 x x'$ and $3 x x'$ .

$\frac{- 9 x^{2} x' - 3 x x' + (- 4 + 3 x) (6 x x' + x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.8

Expand $(- 4 + 3 x) (6 x x' + x')$ using the FOIL Method.

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Step 3.11.6.1.8.1

Apply the distributive property.

$\frac{- 9 x^{2} x' - 3 x x' - 4 (6 x x' + x') + 3 x (6 x x' + x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.8.2

Apply the distributive property.

$\frac{- 9 x^{2} x' - 3 x x' - 4 (6 x x') - 4 x' + 3 x (6 x x' + x')}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.8.3

Apply the distributive property.

$\frac{- 9 x^{2} x' - 3 x x' - 4 (6 x x') - 4 x' + 3 x (6 x x') + 3 x x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.9

Simplify and combine like terms.

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Step 3.11.6.1.9.1

Simplify each term.

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Step 3.11.6.1.9.1.1

Multiply $6$ by $- 4$ .

$\frac{- 9 x^{2} x' - 3 x x' - 24 (x x') - 4 x' + 3 x (6 x x') + 3 x x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.9.1.2

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

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Step 3.11.6.1.9.1.2.1

Move $x$ .

$\frac{- 9 x^{2} x' - 3 x x' - 24 x x' - 4 x' + 3 (x \cdot x) (6 x') + 3 x x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.9.1.2.2

Multiply $x$ by $x$ .

$\frac{- 9 x^{2} x' - 3 x x' - 24 x x' - 4 x' + 3 x^{2} (6 x') + 3 x x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.9.1.3

Multiply $6$ by $3$ .

$\frac{- 9 x^{2} x' - 3 x x' - 24 x x' - 4 x' + 18 x^{2} x' + 3 x x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.1.9.2

Add $- 24 x x'$ and $3 x x'$ .

$\frac{- 9 x^{2} x' - 3 x x' - 21 x x' - 4 x' + 18 x^{2} x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.2

Add $- 9 x^{2} x'$ and $18 x^{2} x'$ .

$\frac{9 x^{2} x' - 3 x x' - 21 x x' - 4 x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.6.3

Subtract $21 x x'$ from $- 3 x x'$ .

$\frac{9 x^{2} x' - 24 x x' - 4 x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.7

Factor $x'$ out of $9 x^{2} x' - 24 x x' - 4 x'$ .

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Step 3.11.7.1

Factor $x'$ out of $9 x^{2} x'$ .

$\frac{x' (9 x^{2}) - 24 x x' - 4 x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.7.2

Factor $x'$ out of $- 24 x x'$ .

$\frac{x' (9 x^{2}) + x' (- 24 x) - 4 x'}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.7.3

Factor $x'$ out of $- 4 x'$ .

$\frac{x' (9 x^{2}) + x' (- 24 x) + x' \cdot - 4}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.7.4

Factor $x'$ out of $x' (9 x^{2}) + x' (- 24 x)$ .

$\frac{x' (9 x^{2} - 24 x) + x' \cdot - 4}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.7.5

Factor $x'$ out of $x' (9 x^{2} - 24 x) + x' \cdot - 4$ .

$\frac{x' (9 x^{2} - 24 x - 4)}{x^{2} {(3 x + 1)}^{2}}$

Step 3.11.8

Reorder factors in $\frac{x' (9 x^{2} - 24 x - 4)}{x^{2} {(3 x + 1)}^{2}}$ .

$\frac{(9 x^{2} - 24 x - 4) x'}{x^{2} {(3 x + 1)}^{2}}$

Step 4

Reform the equation by setting the left side equal to the right side.

$1 = \frac{(9 x^{2} - 24 x - 4) x'}{x^{2} {(3 x + 1)}^{2}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{(9 x^{2} - 24 x - 4) x'}{x^{2} {(3 x + 1)}^{2}} = 1$ .

$\frac{(9 x^{2} - 24 x - 4) x'}{x^{2} {(3 x + 1)}^{2}} = 1$

Step 5.2

Multiply both sides by $x^{2} {(3 x + 1)}^{2}$ .

$\frac{(9 x^{2} - 24 x - 4) x'}{x^{2} {(3 x + 1)}^{2}} (x^{2} {(3 x + 1)}^{2}) = 1 (x^{2} {(3 x + 1)}^{2})$

Step 5.3

Simplify.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{(9 x^{2} - 24 x - 4) x'}{x^{2} {(3 x + 1)}^{2}} (x^{2} {(3 x + 1)}^{2})$ .

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Step 5.3.1.1.1

Cancel the common factor of $x^{2} {(3 x + 1)}^{2}$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{(9 x^{2} - 24 x - 4) x'}{x^{2} {(3 x + 1)}^{2}} (x^{2} {(3 x + 1)}^{2}) = 1 (x^{2} {(3 x + 1)}^{2})$

Step 5.3.1.1.1.2

Rewrite the expression.

$(9 x^{2} - 24 x - 4) x' = 1 (x^{2} {(3 x + 1)}^{2})$

Step 5.3.1.1.2

Apply the distributive property.

$9 x^{2} x' - 24 x x' - 4 x' = 1 (x^{2} {(3 x + 1)}^{2})$

Step 5.3.1.1.3

Reorder.

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Step 5.3.1.1.3.1

Move $x^{2}$ .

$9 x' x^{2} - 24 x x' - 4 x' = 1 (x^{2} {(3 x + 1)}^{2})$

Step 5.3.1.1.3.2

Move $x$ .

$9 x' x^{2} - 24 x' x - 4 x' = 1 (x^{2} {(3 x + 1)}^{2})$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply $x^{2} {(3 x + 1)}^{2}$ by $1$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} {(3 x + 1)}^{2}$

Step 5.4

Solve for $x'$ .

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Step 5.4.1

Simplify $x^{2} {(3 x + 1)}^{2}$ .

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Step 5.4.1.1

Rewrite ${(3 x + 1)}^{2}$ as $(3 x + 1) (3 x + 1)$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} ((3 x + 1) (3 x + 1))$

Step 5.4.1.2

Expand $(3 x + 1) (3 x + 1)$ using the FOIL Method.

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Step 5.4.1.2.1

Apply the distributive property.

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (3 x (3 x + 1) + 1 (3 x + 1))$

Step 5.4.1.2.2

Apply the distributive property.

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (3 x (3 x) + 3 x \cdot 1 + 1 (3 x + 1))$

Step 5.4.1.2.3

Apply the distributive property.

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (3 x (3 x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1)$

Step 5.4.1.3

Simplify and combine like terms.

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Step 5.4.1.3.1

Simplify each term.

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Step 5.4.1.3.1.1

Rewrite using the commutative property of multiplication.

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (3 \cdot 3 x \cdot x + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1)$

Step 5.4.1.3.1.2

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

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Step 5.4.1.3.1.2.1

Move $x$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (3 \cdot 3 (x \cdot x) + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1)$

Step 5.4.1.3.1.2.2

Multiply $x$ by $x$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (3 \cdot 3 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1)$

Step 5.4.1.3.1.3

Multiply $3$ by $3$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (9 x^{2} + 3 x \cdot 1 + 1 (3 x) + 1 \cdot 1)$

Step 5.4.1.3.1.4

Multiply $3$ by $1$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (9 x^{2} + 3 x + 1 (3 x) + 1 \cdot 1)$

Step 5.4.1.3.1.5

Multiply $3 x$ by $1$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (9 x^{2} + 3 x + 3 x + 1 \cdot 1)$

Step 5.4.1.3.1.6

Multiply $1$ by $1$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (9 x^{2} + 3 x + 3 x + 1)$

Step 5.4.1.3.2

Add $3 x$ and $3 x$ .

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (9 x^{2} + 6 x + 1)$

Step 5.4.1.4

Apply the distributive property.

$9 x' x^{2} - 24 x' x - 4 x' = x^{2} (9 x^{2}) + x^{2} (6 x) + x^{2} \cdot 1$

Step 5.4.1.5

Simplify.

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Step 5.4.1.5.1

Rewrite using the commutative property of multiplication.

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{2} x^{2} + x^{2} (6 x) + x^{2} \cdot 1$

Step 5.4.1.5.2

Rewrite using the commutative property of multiplication.

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{2} x^{2} + 6 x^{2} x + x^{2} \cdot 1$

Step 5.4.1.5.3

Multiply $x^{2}$ by $1$ .

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{2} x^{2} + 6 x^{2} x + x^{2}$

Step 5.4.1.6

Simplify each term.

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Step 5.4.1.6.1

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

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Step 5.4.1.6.1.1

Move $x^{2}$ .

$9 x' x^{2} - 24 x' x - 4 x' = 9 (x^{2} x^{2}) + 6 x^{2} x + x^{2}$

Step 5.4.1.6.1.2

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

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{2 + 2} + 6 x^{2} x + x^{2}$

Step 5.4.1.6.1.3

Add $2$ and $2$ .

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{4} + 6 x^{2} x + x^{2}$

Step 5.4.1.6.2

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

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Step 5.4.1.6.2.1

Move $x$ .

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{4} + 6 (x \cdot x^{2}) + x^{2}$

Step 5.4.1.6.2.2

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

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Step 5.4.1.6.2.2.1

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

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{4} + 6 (x^{1} x^{2}) + x^{2}$

Step 5.4.1.6.2.2.2

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

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{4} + 6 x^{1 + 2} + x^{2}$

Step 5.4.1.6.2.3

Add $1$ and $2$ .

$9 x' x^{2} - 24 x' x - 4 x' = 9 x^{4} + 6 x^{3} + x^{2}$

Step 5.4.2

Factor $x'$ out of $9 x' x^{2} - 24 x' x - 4 x'$ .

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Step 5.4.2.1

Factor $x'$ out of $9 x' x^{2}$ .

$x' (9 x^{2}) - 24 x' x - 4 x' = 9 x^{4} + 6 x^{3} + x^{2}$

Step 5.4.2.2

Factor $x'$ out of $- 24 x' x$ .

$x' (9 x^{2}) + x' (- 24 x) - 4 x' = 9 x^{4} + 6 x^{3} + x^{2}$

Step 5.4.2.3

Factor $x'$ out of $- 4 x'$ .

$x' (9 x^{2}) + x' (- 24 x) + x' \cdot - 4 = 9 x^{4} + 6 x^{3} + x^{2}$

Step 5.4.2.4

Factor $x'$ out of $x' (9 x^{2}) + x' (- 24 x)$ .

$x' (9 x^{2} - 24 x) + x' \cdot - 4 = 9 x^{4} + 6 x^{3} + x^{2}$

Step 5.4.2.5

Factor $x'$ out of $x' (9 x^{2} - 24 x) + x' \cdot - 4$ .

$x' (9 x^{2} - 24 x - 4) = 9 x^{4} + 6 x^{3} + x^{2}$

Step 5.4.3

Divide each term in $x' (9 x^{2} - 24 x - 4) = 9 x^{4} + 6 x^{3} + x^{2}$ by $9 x^{2} - 24 x - 4$ and simplify.

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Step 5.4.3.1

Divide each term in $x' (9 x^{2} - 24 x - 4) = 9 x^{4} + 6 x^{3} + x^{2}$ by $9 x^{2} - 24 x - 4$ .

$\frac{x' (9 x^{2} - 24 x - 4)}{9 x^{2} - 24 x - 4} = \frac{9 x^{4}}{9 x^{2} - 24 x - 4} + \frac{6 x^{3}}{9 x^{2} - 24 x - 4} + \frac{x^{2}}{9 x^{2} - 24 x - 4}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $9 x^{2} - 24 x - 4$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{x' (9 x^{2} - 24 x - 4)}{9 x^{2} - 24 x - 4} = \frac{9 x^{4}}{9 x^{2} - 24 x - 4} + \frac{6 x^{3}}{9 x^{2} - 24 x - 4} + \frac{x^{2}}{9 x^{2} - 24 x - 4}$

Step 5.4.3.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{9 x^{4}}{9 x^{2} - 24 x - 4} + \frac{6 x^{3}}{9 x^{2} - 24 x - 4} + \frac{x^{2}}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Combine the numerators over the common denominator.

$x' = \frac{9 x^{4} + 6 x^{3} + x^{2}}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2

Simplify the numerator.

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Step 5.4.3.3.2.1

Factor $x^{2}$ out of $9 x^{4} + 6 x^{3} + x^{2}$ .

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Step 5.4.3.3.2.1.1

Factor $x^{2}$ out of $9 x^{4}$ .

$x' = \frac{x^{2} (9 x^{2}) + 6 x^{3} + x^{2}}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.1.2

Factor $x^{2}$ out of $6 x^{3}$ .

$x' = \frac{x^{2} (9 x^{2}) + x^{2} (6 x) + x^{2}}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.1.3

Multiply by $1$ .

$x' = \frac{x^{2} (9 x^{2}) + x^{2} (6 x) + x^{2} \cdot 1}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.1.4

Factor $x^{2}$ out of $x^{2} (9 x^{2}) + x^{2} (6 x)$ .

$x' = \frac{x^{2} (9 x^{2} + 6 x) + x^{2} \cdot 1}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.1.5

Factor $x^{2}$ out of $x^{2} (9 x^{2} + 6 x) + x^{2} \cdot 1$ .

$x' = \frac{x^{2} (9 x^{2} + 6 x + 1)}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.2

Factor using the perfect square rule.

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Step 5.4.3.3.2.2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

$x' = \frac{x^{2} ({(3 x)}^{2} + 6 x + 1)}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.2.2

Rewrite $1$ as $1^{2}$ .

$x' = \frac{x^{2} ({(3 x)}^{2} + 6 x + 1^{2})}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot (3 x) \cdot 1$

Step 5.4.3.3.2.2.4

Rewrite the polynomial.

$x' = \frac{x^{2} ({(3 x)}^{2} + 2 \cdot (3 x) \cdot 1 + 1^{2})}{9 x^{2} - 24 x - 4}$

Step 5.4.3.3.2.2.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 3 x$ and $b = 1$ .

$x' = \frac{x^{2} {(3 x + 1)}^{2}}{9 x^{2} - 24 x - 4}$

Step 6

Replace $x'$ with $\frac{d x}{d z}$ .

$\frac{d x}{d z} = \frac{x^{2} {(3 x + 1)}^{2}}{9 x^{2} - 24 x - 4}$