Calculus Examples

$y = \frac{4 x^{3}}{2 x^{2} - 5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{4 x^{3}}{2 x^{2} - 5})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $4$ is constant with respect to $y$ , the derivative of $\frac{4 x^{3}}{2 x^{2} - 5}$ with respect to $y$ is $4 \frac{d}{d y} [\frac{x^{3}}{2 x^{2} - 5}]$ .

$4 \frac{d}{d y} [\frac{x^{3}}{2 x^{2} - 5}]$

Step 3.2

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = x^{3}$ and $g (y) = 2 x^{2} - 5$ .

$4 \frac{(2 x^{2} - 5) \frac{d}{d y} [x^{3}] - x^{3} \frac{d}{d y} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{3}$ and $g (y) = x$ .

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Step 3.3.1

To apply the Chain Rule, set $u_{1}$ as $x$ .

$4 \frac{(2 x^{2} - 5) (\frac{d}{d u_{1}} [{u_{1}}^{3}] \frac{d}{d y} [x]) - x^{3} \frac{d}{d y} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{1}} [{u_{1}}^{n}]$ is $n {u_{1}}^{n - 1}$ where $n = 3$ .

$4 \frac{(2 x^{2} - 5) (3 {u_{1}}^{2} \frac{d}{d y} [x]) - x^{3} \frac{d}{d y} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.3.3

Replace all occurrences of $u_{1}$ with $x$ .

$4 \frac{(2 x^{2} - 5) (3 x^{2} \frac{d}{d y} [x]) - x^{3} \frac{d}{d y} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.4

Move $3$ to the left of $2 x^{2} - 5$ .

$4 \frac{3 \cdot (2 x^{2} - 5) x^{2} \frac{d}{d y} [x] - x^{3} \frac{d}{d y} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.5

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

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} \frac{d}{d y} [2 x^{2} - 5]}{{(2 x^{2} - 5)}^{2}}$

Step 3.6

Differentiate.

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Step 3.6.1

By the Sum Rule, the derivative of $2 x^{2} - 5$ with respect to $y$ is $\frac{d}{d y} [2 x^{2}] + \frac{d}{d y} [- 5]$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (\frac{d}{d y} [2 x^{2}] + \frac{d}{d y} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.6.2

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

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (2 \frac{d}{d y} [x^{2}] + \frac{d}{d y} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.7

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

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Step 3.7.1

To apply the Chain Rule, set $u_{2}$ as $x$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (2 (\frac{d}{d u_{2}} [{u_{2}}^{2}] \frac{d}{d y} [x]) + \frac{d}{d y} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.7.2

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

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (2 (2 u_{2} \frac{d}{d y} [x]) + \frac{d}{d y} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.7.3

Replace all occurrences of $u_{2}$ with $x$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (2 (2 x \frac{d}{d y} [x]) + \frac{d}{d y} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.8

Multiply $2$ by $2$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (4 (x \frac{d}{d y} [x]) + \frac{d}{d y} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.9

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

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (4 x x' + \frac{d}{d y} [- 5])}{{(2 x^{2} - 5)}^{2}}$

Step 3.10

Since $- 5$ is constant with respect to $y$ , the derivative of $- 5$ with respect to $y$ is $0$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (4 x x' + 0)}{{(2 x^{2} - 5)}^{2}}$

Step 3.11

Simplify the expression.

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Step 3.11.1

Add $4 x x'$ and $0$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - x^{3} (4 x x')}{{(2 x^{2} - 5)}^{2}}$

Step 3.11.2

Multiply $4$ by $- 1$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - 4 x^{3} (x x')}{{(2 x^{2} - 5)}^{2}}$

Step 3.12

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

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - 4 (x^{1} x^{3}) x'}{{(2 x^{2} - 5)}^{2}}$

Step 3.13

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

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

Step 3.14

Add $1$ and $3$ .

$4 \frac{3 (2 x^{2} - 5) x^{2} x' - 4 x^{4} x'}{{(2 x^{2} - 5)}^{2}}$

Step 3.15

Combine $4$ and $\frac{3 (2 x^{2} - 5) x^{2} x' - 4 x^{4} x'}{{(2 x^{2} - 5)}^{2}}$ .

$\frac{4 (3 (2 x^{2} - 5) x^{2} x' - 4 x^{4} x')}{{(2 x^{2} - 5)}^{2}}$

Step 3.16

Simplify.

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Step 3.16.1

Apply the distributive property.

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

Step 3.16.2

Apply the distributive property.

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

Step 3.16.3

Apply the distributive property.

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

Step 3.16.4

Apply the distributive property.

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

Step 3.16.5

Simplify the numerator.

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Step 3.16.5.1

Simplify each term.

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Step 3.16.5.1.1

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

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Step 3.16.5.1.1.1

Move $x^{2}$ .

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

Step 3.16.5.1.1.2

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

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

Step 3.16.5.1.1.3

Add $2$ and $2$ .

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

Step 3.16.5.1.2

Multiply $2$ by $3$ .

$\frac{4 (6 x^{4} x') + 4 (3 \cdot - 5 x^{2} x') + 4 (- 4 x^{4} x')}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.5.1.3

Multiply $6$ by $4$ .

$\frac{24 (x^{4} x') + 4 (3 \cdot - 5 x^{2} x') + 4 (- 4 x^{4} x')}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.5.1.4

Multiply $3$ by $- 5$ .

$\frac{24 x^{4} x' + 4 (- 15 x^{2} x') + 4 (- 4 x^{4} x')}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.5.1.5

Multiply $- 15$ by $4$ .

$\frac{24 x^{4} x' - 60 (x^{2} x') + 4 (- 4 x^{4} x')}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.5.1.6

Multiply $- 4$ by $4$ .

$\frac{24 x^{4} x' - 60 x^{2} x' - 16 x^{4} x'}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.5.2

Subtract $16 x^{4} x'$ from $24 x^{4} x'$ .

$\frac{8 x^{4} x' - 60 x^{2} x'}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.6

Factor $4 x^{2} x'$ out of $8 x^{4} x' - 60 x^{2} x'$ .

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Step 3.16.6.1

Factor $4 x^{2} x'$ out of $8 x^{4} x'$ .

$\frac{4 x^{2} x' (2 x^{2}) - 60 x^{2} x'}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.6.2

Factor $4 x^{2} x'$ out of $- 60 x^{2} x'$ .

$\frac{4 x^{2} x' (2 x^{2}) + 4 x^{2} x' (- 15)}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.6.3

Factor $4 x^{2} x'$ out of $4 x^{2} x' (2 x^{2}) + 4 x^{2} x' (- 15)$ .

$\frac{4 x^{2} x' (2 x^{2} - 15)}{{(2 x^{2} - 5)}^{2}}$

Step 3.16.7

Reorder factors in $\frac{4 x^{2} x' (2 x^{2} - 15)}{{(2 x^{2} - 5)}^{2}}$ .

$\frac{4 x^{2} (2 x^{2} - 15) x'}{{(2 x^{2} - 5)}^{2}}$

Step 4

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

$1 = \frac{4 x^{2} (2 x^{2} - 15) x'}{{(2 x^{2} - 5)}^{2}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{4 x^{2} (2 x^{2} - 15) x'}{{(2 x^{2} - 5)}^{2}} = 1$ .

$\frac{4 x^{2} (2 x^{2} - 15) x'}{{(2 x^{2} - 5)}^{2}} = 1$

Step 5.2

Multiply both sides by ${(2 x^{2} - 5)}^{2}$ .

$\frac{4 x^{2} (2 x^{2} - 15) x'}{{(2 x^{2} - 5)}^{2}} {(2 x^{2} - 5)}^{2} = 1 {(2 x^{2} - 5)}^{2}$

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 $\frac{4 x^{2} (2 x^{2} - 15) x'}{{(2 x^{2} - 5)}^{2}} {(2 x^{2} - 5)}^{2}$ .

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Step 5.3.1.1.1

Simplify terms.

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Step 5.3.1.1.1.1

Cancel the common factor of ${(2 x^{2} - 5)}^{2}$ .

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Step 5.3.1.1.1.1.1

Cancel the common factor.

$\frac{4 x^{2} (2 x^{2} - 15) x'}{{(2 x^{2} - 5)}^{2}} {(2 x^{2} - 5)}^{2} = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.1.1.2

Rewrite the expression.

$4 x^{2} (2 x^{2} - 15) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.1.2

Apply the distributive property.

$(4 x^{2} (2 x^{2}) + 4 x^{2} \cdot - 15) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.1.3

Simplify the expression.

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Step 5.3.1.1.1.3.1

Rewrite using the commutative property of multiplication.

$(4 \cdot 2 x^{2} x^{2} + 4 x^{2} \cdot - 15) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.1.3.2

Multiply $- 15$ by $4$ .

$(4 \cdot 2 x^{2} x^{2} - 60 x^{2}) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.2

Simplify each term.

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Step 5.3.1.1.2.1

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

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Step 5.3.1.1.2.1.1

Move $x^{2}$ .

$(4 \cdot 2 (x^{2} x^{2}) - 60 x^{2}) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.2.1.2

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

$(4 \cdot 2 x^{2 + 2} - 60 x^{2}) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.2.1.3

Add $2$ and $2$ .

$(4 \cdot 2 x^{4} - 60 x^{2}) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.2.2

Multiply $4$ by $2$ .

$(8 x^{4} - 60 x^{2}) x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.3

Simplify by multiplying through.

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Step 5.3.1.1.3.1

Apply the distributive property.

$8 x^{4} x' - 60 x^{2} x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.3.2

Reorder.

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Step 5.3.1.1.3.2.1

Move $x^{4}$ .

$8 x' x^{4} - 60 x^{2} x' = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.1.1.3.2.2

Move $x^{2}$ .

$8 x' x^{4} - 60 x' x^{2} = 1 {(2 x^{2} - 5)}^{2}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Multiply ${(2 x^{2} - 5)}^{2}$ by $1$ .

$8 x' x^{4} - 60 x' x^{2} = {(2 x^{2} - 5)}^{2}$

Step 5.4

Solve for $x'$ .

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Step 5.4.1

Simplify ${(2 x^{2} - 5)}^{2}$ .

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Step 5.4.1.1

Rewrite ${(2 x^{2} - 5)}^{2}$ as $(2 x^{2} - 5) (2 x^{2} - 5)$ .

$8 x' x^{4} - 60 x' x^{2} = (2 x^{2} - 5) (2 x^{2} - 5)$

Step 5.4.1.2

Expand $(2 x^{2} - 5) (2 x^{2} - 5)$ using the FOIL Method.

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Step 5.4.1.2.1

Apply the distributive property.

$8 x' x^{4} - 60 x' x^{2} = 2 x^{2} (2 x^{2} - 5) - 5 (2 x^{2} - 5)$

Step 5.4.1.2.2

Apply the distributive property.

$8 x' x^{4} - 60 x' x^{2} = 2 x^{2} (2 x^{2}) + 2 x^{2} \cdot - 5 - 5 (2 x^{2} - 5)$

Step 5.4.1.2.3

Apply the distributive property.

$8 x' x^{4} - 60 x' x^{2} = 2 x^{2} (2 x^{2}) + 2 x^{2} \cdot - 5 - 5 (2 x^{2}) - 5 \cdot - 5$

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.

$8 x' x^{4} - 60 x' x^{2} = 2 \cdot 2 x^{2} x^{2} + 2 x^{2} \cdot - 5 - 5 (2 x^{2}) - 5 \cdot - 5$

Step 5.4.1.3.1.2

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

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Step 5.4.1.3.1.2.1

Move $x^{2}$ .

$8 x' x^{4} - 60 x' x^{2} = 2 \cdot 2 (x^{2} x^{2}) + 2 x^{2} \cdot - 5 - 5 (2 x^{2}) - 5 \cdot - 5$

Step 5.4.1.3.1.2.2

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

$8 x' x^{4} - 60 x' x^{2} = 2 \cdot 2 x^{2 + 2} + 2 x^{2} \cdot - 5 - 5 (2 x^{2}) - 5 \cdot - 5$

Step 5.4.1.3.1.2.3

Add $2$ and $2$ .

$8 x' x^{4} - 60 x' x^{2} = 2 \cdot 2 x^{4} + 2 x^{2} \cdot - 5 - 5 (2 x^{2}) - 5 \cdot - 5$

Step 5.4.1.3.1.3

Multiply $2$ by $2$ .

$8 x' x^{4} - 60 x' x^{2} = 4 x^{4} + 2 x^{2} \cdot - 5 - 5 (2 x^{2}) - 5 \cdot - 5$

Step 5.4.1.3.1.4

Multiply $- 5$ by $2$ .

$8 x' x^{4} - 60 x' x^{2} = 4 x^{4} - 10 x^{2} - 5 (2 x^{2}) - 5 \cdot - 5$

Step 5.4.1.3.1.5

Multiply $2$ by $- 5$ .

$8 x' x^{4} - 60 x' x^{2} = 4 x^{4} - 10 x^{2} - 10 x^{2} - 5 \cdot - 5$

Step 5.4.1.3.1.6

Multiply $- 5$ by $- 5$ .

$8 x' x^{4} - 60 x' x^{2} = 4 x^{4} - 10 x^{2} - 10 x^{2} + 25$

Step 5.4.1.3.2

Subtract $10 x^{2}$ from $- 10 x^{2}$ .

$8 x' x^{4} - 60 x' x^{2} = 4 x^{4} - 20 x^{2} + 25$

Step 5.4.2

Factor $4 x' x^{2}$ out of $8 x' x^{4} - 60 x' x^{2}$ .

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Step 5.4.2.1

Factor $4 x' x^{2}$ out of $8 x' x^{4}$ .

$4 x' x^{2} (2 x^{2}) - 60 x' x^{2} = 4 x^{4} - 20 x^{2} + 25$

Step 5.4.2.2

Factor $4 x' x^{2}$ out of $- 60 x' x^{2}$ .

$4 x' x^{2} (2 x^{2}) + 4 x' x^{2} \cdot - 15 = 4 x^{4} - 20 x^{2} + 25$

Step 5.4.2.3

Factor $4 x' x^{2}$ out of $4 x' x^{2} (2 x^{2}) + 4 x' x^{2} \cdot - 15$ .

$4 x' x^{2} (2 x^{2} - 15) = 4 x^{4} - 20 x^{2} + 25$

Step 5.4.3

Divide each term in $4 x' x^{2} (2 x^{2} - 15) = 4 x^{4} - 20 x^{2} + 25$ by $4 x^{2} (2 x^{2} - 15)$ and simplify.

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Step 5.4.3.1

Divide each term in $4 x' x^{2} (2 x^{2} - 15) = 4 x^{4} - 20 x^{2} + 25$ by $4 x^{2} (2 x^{2} - 15)$ .

$\frac{4 x' x^{2} (2 x^{2} - 15)}{4 x^{2} (2 x^{2} - 15)} = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.2

Simplify the left side.

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Step 5.4.3.2.1

Cancel the common factor of $4$ .

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Step 5.4.3.2.1.1

Cancel the common factor.

$\frac{4 x' x^{2} (2 x^{2} - 15)}{4 x^{2} (2 x^{2} - 15)} = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.2.1.2

Rewrite the expression.

$\frac{x' x^{2} (2 x^{2} - 15)}{x^{2} (2 x^{2} - 15)} = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.2.2

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

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Step 5.4.3.2.2.1

Cancel the common factor.

$\frac{x' x^{2} (2 x^{2} - 15)}{x^{2} (2 x^{2} - 15)} = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.2.2.2

Rewrite the expression.

$\frac{x' (2 x^{2} - 15)}{2 x^{2} - 15} = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.2.3

Cancel the common factor of $2 x^{2} - 15$ .

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Step 5.4.3.2.3.1

Cancel the common factor.

$\frac{x' (2 x^{2} - 15)}{2 x^{2} - 15} = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.2.3.2

Divide $x'$ by $1$ .

$x' = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3

Simplify the right side.

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Step 5.4.3.3.1

Simplify each term.

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Step 5.4.3.3.1.1

Cancel the common factor of $4$ .

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Step 5.4.3.3.1.1.1

Cancel the common factor.

$x' = \frac{4 x^{4}}{4 x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.1.2

Rewrite the expression.

$x' = \frac{x^{4}}{x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.2

Cancel the common factor of $x^{4}$ and $x^{2}$ .

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Step 5.4.3.3.1.2.1

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

$x' = \frac{x^{2} x^{2}}{x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.2.2

Cancel the common factors.

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Step 5.4.3.3.1.2.2.1

Cancel the common factor.

$x' = \frac{x^{2} x^{2}}{x^{2} (2 x^{2} - 15)} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.2.2.2

Rewrite the expression.

$x' = \frac{x^{2}}{2 x^{2} - 15} + \frac{- 20 x^{2}}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.3

Cancel the common factor of $- 20$ and $4$ .

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Step 5.4.3.3.1.3.1

Factor $4$ out of $- 20 x^{2}$ .

$x' = \frac{x^{2}}{2 x^{2} - 15} + \frac{4 (- 5 x^{2})}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.3.2

Cancel the common factors.

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Step 5.4.3.3.1.3.2.1

Factor $4$ out of $4 x^{2} (2 x^{2} - 15)$ .

$x' = \frac{x^{2}}{2 x^{2} - 15} + \frac{4 (- 5 x^{2})}{4 (x^{2} (2 x^{2} - 15))} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.3.2.2

Cancel the common factor.

$x' = \frac{x^{2}}{2 x^{2} - 15} + \frac{4 (- 5 x^{2})}{4 (x^{2} (2 x^{2} - 15))} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.3.2.3

Rewrite the expression.

$x' = \frac{x^{2}}{2 x^{2} - 15} + \frac{- 5 x^{2}}{x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.4

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

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Step 5.4.3.3.1.4.1

Cancel the common factor.

$x' = \frac{x^{2}}{2 x^{2} - 15} + \frac{- 5 x^{2}}{x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.4.2

Rewrite the expression.

$x' = \frac{x^{2}}{2 x^{2} - 15} + \frac{- 5}{2 x^{2} - 15} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.1.5

Move the negative in front of the fraction.

$x' = \frac{x^{2}}{2 x^{2} - 15} - \frac{5}{2 x^{2} - 15} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.2

Combine the numerators over the common denominator.

$x' = \frac{x^{2} - 5}{2 x^{2} - 15} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.3

To write $\frac{x^{2} - 5}{2 x^{2} - 15}$ as a fraction with a common denominator, multiply by $\frac{4 x^{2}}{4 x^{2}}$ .

$x' = \frac{x^{2} - 5}{2 x^{2} - 15} \cdot \frac{4 x^{2}}{4 x^{2}} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.4

Write each expression with a common denominator of $(2 x^{2} - 15) \cdot 4 x^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.4.3.3.4.1

Multiply $\frac{x^{2} - 5}{2 x^{2} - 15}$ by $\frac{4 x^{2}}{4 x^{2}}$ .

$x' = \frac{(x^{2} - 5) (4 x^{2})}{(2 x^{2} - 15) (4 x^{2})} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.4.2

Reorder the factors of $(2 x^{2} - 15) (4 x^{2})$ .

$x' = \frac{(x^{2} - 5) (4 x^{2})}{4 x^{2} (2 x^{2} - 15)} + \frac{25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.5

Combine the numerators over the common denominator.

$x' = \frac{(x^{2} - 5) (4 x^{2}) + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6

Simplify the numerator.

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Step 5.4.3.3.6.1

Apply the distributive property.

$x' = \frac{(x^{2} \cdot 4 - 5 \cdot 4) x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.2

Move $4$ to the left of $x^{2}$ .

$x' = \frac{(4 \cdot x^{2} - 5 \cdot 4) x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.3

Multiply $- 5$ by $4$ .

$x' = \frac{(4 \cdot x^{2} - 20) x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.4

Apply the distributive property.

$x' = \frac{4 x^{2} x^{2} - 20 x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.5

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

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Step 5.4.3.3.6.5.1

Move $x^{2}$ .

$x' = \frac{4 (x^{2} x^{2}) - 20 x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.5.2

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

$x' = \frac{4 x^{2 + 2} - 20 x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.5.3

Add $2$ and $2$ .

$x' = \frac{4 x^{4} - 20 x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.6

Rewrite $4 x^{4} - 20 x^{2} + 25$ in a factored form.

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Step 5.4.3.3.6.6.1

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

$x' = \frac{4 {(x^{2})}^{2} - 20 x^{2} + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.6.2

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

$x' = \frac{4 u^{2} - 20 u + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.6.3

Factor using the perfect square rule.

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Step 5.4.3.3.6.6.3.1

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

$x' = \frac{{(2 u)}^{2} - 20 u + 25}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.6.3.2

Rewrite $25$ as $5^{2}$ .

$x' = \frac{{(2 u)}^{2} - 20 u + 5^{2}}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.6.3.3

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

$20 u = 2 \cdot (2 u) \cdot 5$

Step 5.4.3.3.6.6.3.4

Rewrite the polynomial.

$x' = \frac{{(2 u)}^{2} - 2 \cdot (2 u) \cdot 5 + 5^{2}}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.6.3.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 u$ and $b = 5$ .

$x' = \frac{{(2 u - 5)}^{2}}{4 x^{2} (2 x^{2} - 15)}$

Step 5.4.3.3.6.6.4

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

$x' = \frac{{(2 x^{2} - 5)}^{2}}{4 x^{2} (2 x^{2} - 15)}$

Step 6

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

$\frac{d x}{d y} = \frac{{(2 x^{2} - 5)}^{2}}{4 x^{2} (2 x^{2} - 15)}$