Calculus Examples

$y = \sqrt{x} + x^{\frac{3}{2}} - 8 x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$y = x^{\frac{1}{2}} + x^{\frac{3}{2}} - 8 x$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (x^{\frac{1}{2}} + x^{\frac{3}{2}} - 8 x)$

Step 3

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

$1$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $x^{\frac{1}{2}} + x^{\frac{3}{2}} - 8 x$ with respect to $y$ is $\frac{d}{d y} [x^{\frac{1}{2}}] + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$ .

$\frac{d}{d y} [x^{\frac{1}{2}}] + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2

Evaluate $\frac{d}{d y} [x^{\frac{1}{2}}]$ .

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Step 4.2.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{1}{2}}$ and $g (y) = x$ .

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Step 4.2.1.1

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

$\frac{d}{d u_{1}} [{u_{1}}^{\frac{1}{2}}] \frac{d}{d y} [x] + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.1.2

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

$\frac{1}{2} {u_{1}}^{\frac{1}{2} - 1} \frac{d}{d y} [x] + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.1.3

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

$\frac{1}{2} x^{\frac{1}{2} - 1} \frac{d}{d y} [x] + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.2

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

$\frac{1}{2} x^{\frac{1}{2} - 1} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.6

Simplify the numerator.

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Step 4.2.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{\frac{1 - 2}{2}} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{\frac{- 1}{2}} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.8

Combine $\frac{1}{2}$ and $x^{- \frac{1}{2}}$ .

$\frac{x^{- \frac{1}{2}}}{2} x' + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.9

Combine $\frac{x^{- \frac{1}{2}}}{2}$ and $x'$ .

$\frac{x^{- \frac{1}{2}} x'}{2} + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.2.10

Move $x^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{d}{d y} [x^{\frac{3}{2}}] + \frac{d}{d y} [- 8 x]$

Step 4.3

Evaluate $\frac{d}{d y} [x^{\frac{3}{2}}]$ .

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Step 4.3.1

Differentiate using the chain rule, which states that $\frac{d}{d y} [f (g (y))]$ is $f' (g (y)) g' (y)$ where $f (y) = y^{\frac{3}{2}}$ and $g (y) = x$ .

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Step 4.3.1.1

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

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{d}{d u_{2}} [{u_{2}}^{\frac{3}{2}}] \frac{d}{d y} [x] + \frac{d}{d y} [- 8 x]$

Step 4.3.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = \frac{3}{2}$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} {u_{2}}^{\frac{3}{2} - 1} \frac{d}{d y} [x] + \frac{d}{d y} [- 8 x]$

Step 4.3.1.3

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

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} - 1} \frac{d}{d y} [x] + \frac{d}{d y} [- 8 x]$

Step 4.3.2

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

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} - 1} x' + \frac{d}{d y} [- 8 x]$

Step 4.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}} x' + \frac{d}{d y} [- 8 x]$

Step 4.3.4

Combine $- 1$ and $\frac{2}{2}$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}} x' + \frac{d}{d y} [- 8 x]$

Step 4.3.5

Combine the numerators over the common denominator.

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}} x' + \frac{d}{d y} [- 8 x]$

Step 4.3.6

Simplify the numerator.

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Step 4.3.6.1

Multiply $- 1$ by $2$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{3 - 2}{2}} x' + \frac{d}{d y} [- 8 x]$

Step 4.3.6.2

Subtract $2$ from $3$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3}{2} x^{\frac{1}{2}} x' + \frac{d}{d y} [- 8 x]$

Step 4.3.7

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}}}{2} x' + \frac{d}{d y} [- 8 x]$

Step 4.3.8

Combine $\frac{3 x^{\frac{1}{2}}}{2}$ and $x'$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} + \frac{d}{d y} [- 8 x]$

Step 4.4

Evaluate $\frac{d}{d y} [- 8 x]$ .

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Step 4.4.1

Since $- 8$ is constant with respect to $y$ , the derivative of $- 8 x$ with respect to $y$ is $- 8 \frac{d}{d y} [x]$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} - 8 \frac{d}{d y} [x]$

Step 4.4.2

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

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} - 8 x'$

Step 5

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

$1 = \frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} - 8 x'$

Step 6

Solve for $x^{\frac{1}{2}}$ .

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Step 6.1

Rewrite the equation as $\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} - 8 x' = 1$ .

$\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} - 8 x' = 1$

Step 6.2

Find the LCD of the terms in the equation.

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Step 6.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 x^{\frac{1}{2}}, 2, 1, 1$

Step 6.2.2

Since $2 x^{\frac{1}{2}}, 2, 1, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $2, 2, 1, 1$ then find LCM for the variable part $x^{\frac{1}{2}}$ .

Step 6.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 6.2.4

Since $2$ has no factors besides $1$ and $2$ .

$2$ is a prime number

Step 6.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 6.2.6

The LCM of $2, 2, 1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2$

Step 6.2.7

The LCM of $x^{\frac{1}{2}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x^{\frac{1}{2}}$

Step 6.2.8

The LCM for $2 x^{\frac{1}{2}}, 2, 1, 1$ is the numeric part $2$ multiplied by the variable part.

$2 x^{\frac{1}{2}}$

Step 6.3

Multiply each term in $\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} - 8 x' = 1$ by $2 x^{\frac{1}{2}}$ to eliminate the fractions.

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Step 6.3.1

Multiply each term in $\frac{x'}{2 x^{\frac{1}{2}}} + \frac{3 x^{\frac{1}{2}} x'}{2} - 8 x' = 1$ by $2 x^{\frac{1}{2}}$ .

$\frac{x'}{2 x^{\frac{1}{2}}} (2 x^{\frac{1}{2}}) + \frac{3 x^{\frac{1}{2}} x'}{2} (2 x^{\frac{1}{2}}) - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Simplify each term.

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Step 6.3.2.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{x'}{2 x^{\frac{1}{2}}} x^{\frac{1}{2}} + \frac{3 x^{\frac{1}{2}} x'}{2} (2 x^{\frac{1}{2}}) - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.2

Cancel the common factor of $2$ .

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Step 6.3.2.1.2.1

Cancel the common factor.

$2 \frac{x'}{2 x^{\frac{1}{2}}} x^{\frac{1}{2}} + \frac{3 x^{\frac{1}{2}} x'}{2} (2 x^{\frac{1}{2}}) - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.2.2

Rewrite the expression.

$\frac{x'}{x^{\frac{1}{2}}} x^{\frac{1}{2}} + \frac{3 x^{\frac{1}{2}} x'}{2} (2 x^{\frac{1}{2}}) - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.3

Cancel the common factor of $x^{\frac{1}{2}}$ .

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Step 6.3.2.1.3.1

Cancel the common factor.

$\frac{x'}{x^{\frac{1}{2}}} x^{\frac{1}{2}} + \frac{3 x^{\frac{1}{2}} x'}{2} (2 x^{\frac{1}{2}}) - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.3.2

Rewrite the expression.

$x' + \frac{3 x^{\frac{1}{2}} x'}{2} (2 x^{\frac{1}{2}}) - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.4

Rewrite using the commutative property of multiplication.

$x' + 2 \frac{3 x^{\frac{1}{2}} x'}{2} x^{\frac{1}{2}} - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.5

Cancel the common factor of $2$ .

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Step 6.3.2.1.5.1

Cancel the common factor.

$x' + 2 \frac{3 x^{\frac{1}{2}} x'}{2} x^{\frac{1}{2}} - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.5.2

Rewrite the expression.

$x' + 3 x^{\frac{1}{2}} x' x^{\frac{1}{2}} - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.6

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{1}{2}}$ by adding the exponents.

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Step 6.3.2.1.6.1

Move $x^{\frac{1}{2}}$ .

$x' + 3 (x^{\frac{1}{2}} x^{\frac{1}{2}}) x' - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.6.2

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

$x' + 3 x^{\frac{1}{2} + \frac{1}{2}} x' - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.6.3

Combine the numerators over the common denominator.

$x' + 3 x^{\frac{1 + 1}{2}} x' - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.6.4

Add $1$ and $1$ .

$x' + 3 x^{\frac{2}{2}} x' - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.6.5

Divide $2$ by $2$ .

$x' + 3 x^{1} x' - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.7

Simplify $3 x^{1} x'$ .

$x' + 3 x x' - 8 x' (2 x^{\frac{1}{2}}) = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.8

Rewrite using the commutative property of multiplication.

$x' + 3 x x' - 8 \cdot 2 x' x^{\frac{1}{2}} = 1 (2 x^{\frac{1}{2}})$

Step 6.3.2.1.9

Multiply $- 8$ by $2$ .

$x' + 3 x x' - 16 x' x^{\frac{1}{2}} = 1 (2 x^{\frac{1}{2}})$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

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

$x' + 3 x x' - 16 x' x^{\frac{1}{2}} = 2 x^{\frac{1}{2}}$

Step 6.4

Solve the equation.

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Step 6.4.1

Find a common factor $x^{\frac{1}{2}}$ that is present in each term.

${({x'}^{\frac{1}{2}})}^{2} + 3 {(x {x'}^{\frac{1}{2}})}^{2} - 16 x' x^{\frac{1}{2}} - 2 x^{\frac{1}{2}}$

Step 6.4.2

Substitute $u$ for $x^{\frac{1}{2}}$ .

${({x'}^{\frac{1}{2}})}^{2} + 3 {(x {x'}^{\frac{1}{2}})}^{2} - 16 x' u - 2 u = 0$

Step 6.4.3

Solve for $u$ .

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Step 6.4.3.1

Simplify each term.

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Step 6.4.3.1.1

Multiply the exponents in ${({x'}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.3.1.1.1

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

${x'}^{\frac{1}{2} \cdot 2} + 3 {(x {x'}^{\frac{1}{2}})}^{2} - 16 x' u - 2 u = 0$

Step 6.4.3.1.1.2

Cancel the common factor of $2$ .

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Step 6.4.3.1.1.2.1

Cancel the common factor.

${x'}^{\frac{1}{2} \cdot 2} + 3 {(x {x'}^{\frac{1}{2}})}^{2} - 16 x' u - 2 u = 0$

Step 6.4.3.1.1.2.2

Rewrite the expression.

${x'}^{1} + 3 {(x {x'}^{\frac{1}{2}})}^{2} - 16 x' u - 2 u = 0$

Step 6.4.3.1.2

Simplify.

$x' + 3 {(x {x'}^{\frac{1}{2}})}^{2} - 16 x' u - 2 u = 0$

Step 6.4.3.1.3

Apply the product rule to $x {x'}^{\frac{1}{2}}$ .

$x' + 3 (x^{2} {({x'}^{\frac{1}{2}})}^{2}) - 16 x' u - 2 u = 0$

Step 6.4.3.1.4

Multiply the exponents in ${({x'}^{\frac{1}{2}})}^{2}$ .

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Step 6.4.3.1.4.1

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

$x' + 3 (x^{2} {x'}^{\frac{1}{2} \cdot 2}) - 16 x' u - 2 u = 0$

Step 6.4.3.1.4.2

Cancel the common factor of $2$ .

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Step 6.4.3.1.4.2.1

Cancel the common factor.

$x' + 3 (x^{2} {x'}^{\frac{1}{2} \cdot 2}) - 16 x' u - 2 u = 0$

Step 6.4.3.1.4.2.2

Rewrite the expression.

$x' + 3 (x^{2} {x'}^{1}) - 16 x' u - 2 u = 0$

Step 6.4.3.1.5

Simplify.

$x' + 3 x^{2} x' - 16 x' u - 2 u = 0$

Step 6.4.3.2

Move all terms not containing $u$ to the right side of the equation.

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Step 6.4.3.2.1

Subtract $x'$ from both sides of the equation.

$3 x^{2} x' - 16 x' u - 2 u = - x'$

Step 6.4.3.2.2

Subtract $3 x^{2} x'$ from both sides of the equation.

$- 16 x' u - 2 u = - x' - 3 x^{2} x'$

Step 6.4.3.3

Factor $2 u$ out of $- 16 x' u - 2 u$ .

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Step 6.4.3.3.1

Factor $2 u$ out of $- 16 x' u$ .

$2 u (- 8 x') - 2 u = - x' - 3 x^{2} x'$

Step 6.4.3.3.2

Factor $2 u$ out of $- 2 u$ .

$2 u (- 8 x') + 2 u (- 1) = - x' - 3 x^{2} x'$

Step 6.4.3.3.3

Factor $2 u$ out of $2 u (- 8 x') + 2 u (- 1)$ .

$2 u (- 8 x' - 1) = - x' - 3 x^{2} x'$

Step 6.4.3.4

Divide each term in $2 u (- 8 x' - 1) = - x' - 3 x^{2} x'$ by $2 (- 8 x' - 1)$ and simplify.

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Step 6.4.3.4.1

Divide each term in $2 u (- 8 x' - 1) = - x' - 3 x^{2} x'$ by $2 (- 8 x' - 1)$ .

$\frac{2 u (- 8 x' - 1)}{2 (- 8 x' - 1)} = \frac{- x'}{2 (- 8 x' - 1)} + \frac{- 3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 6.4.3.4.2

Simplify the left side.

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Step 6.4.3.4.2.1

Cancel the common factor of $2$ .

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Step 6.4.3.4.2.1.1

Cancel the common factor.

$\frac{2 u (- 8 x' - 1)}{2 (- 8 x' - 1)} = \frac{- x'}{2 (- 8 x' - 1)} + \frac{- 3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 6.4.3.4.2.1.2

Rewrite the expression.

$\frac{u (- 8 x' - 1)}{- 8 x' - 1} = \frac{- x'}{2 (- 8 x' - 1)} + \frac{- 3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 6.4.3.4.2.2

Cancel the common factor of $- 8 x' - 1$ .

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Step 6.4.3.4.2.2.1

Cancel the common factor.

$\frac{u (- 8 x' - 1)}{- 8 x' - 1} = \frac{- x'}{2 (- 8 x' - 1)} + \frac{- 3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 6.4.3.4.2.2.2

Divide $u$ by $1$ .

$u = \frac{- x'}{2 (- 8 x' - 1)} + \frac{- 3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 6.4.3.4.3

Simplify the right side.

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Step 6.4.3.4.3.1

Simplify each term.

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Step 6.4.3.4.3.1.1

Move the negative in front of the fraction.

$u = - \frac{x'}{2 (- 8 x' - 1)} + \frac{- 3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 6.4.3.4.3.1.2

Move the negative in front of the fraction.

$u = - \frac{x'}{2 (- 8 x' - 1)} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 6.4.4

Substitute $x'$ for $u$ .

$x^{\frac{1}{2}} = - \frac{x'}{2 (- 8 x' - 1)} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7

Simplify each term.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Factor $- 1$ out of $- 8 x' - 1$ .

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Step 7.1.1.1

Factor $- 1$ out of $- 8 x'$ .

$x^{\frac{1}{2}} = - \frac{x'}{2 (- (8 x') - 1)} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.1.1.2

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

$x^{\frac{1}{2}} = - \frac{x'}{2 (- (8 x') - 1 \cdot 1)} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.1.1.3

Factor $- 1$ out of $- (8 x') - 1 (1)$ .

$x^{\frac{1}{2}} = - \frac{x'}{2 (- (8 x' + 1))} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

$x^{\frac{1}{2}} = - \frac{x'}{2 \cdot (- 1 (8 x' + 1))} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.1.2

Combine exponents.

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Step 7.1.2.1

Factor out negative.

$x^{\frac{1}{2}} = - \frac{x'}{- (2 (8 x' + 1))} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.1.2.2

Multiply $2$ by $- 1$ .

$x^{\frac{1}{2}} = - \frac{x'}{- 2 (8 x' + 1)} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.2

Move the negative in front of the fraction.

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.3

Multiply $- - \frac{x'}{2 (8 x' + 1)}$ .

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Step 7.3.1

Multiply $- 1$ by $- 1$ .

$x^{\frac{1}{2}} = 1 (\frac{x'}{2 (8 x' + 1)}) - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.3.2

Multiply $\frac{x'}{2 (8 x' + 1)}$ by $1$ .

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{2 (- 8 x' - 1)}$

Step 7.4

Simplify the denominator.

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Step 7.4.1

Factor $- 1$ out of $- 8 x' - 1$ .

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Step 7.4.1.1

Factor $- 1$ out of $- 8 x'$ .

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{2 (- (8 x') - 1)}$

Step 7.4.1.2

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

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{2 (- (8 x') - 1 \cdot 1)}$

Step 7.4.1.3

Factor $- 1$ out of $- (8 x') - 1 (1)$ .

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{2 (- (8 x' + 1))}$

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{2 \cdot (- 1 (8 x' + 1))}$

Step 7.4.2

Combine exponents.

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Step 7.4.2.1

Factor out negative.

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{- (2 (8 x' + 1))}$

Step 7.4.2.2

Multiply $2$ by $- 1$ .

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} - \frac{3 x^{2} x'}{- 2 (8 x' + 1)}$

Step 7.5

Move the negative in front of the fraction.

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} + \frac{3 x^{2} x'}{2 (8 x' + 1)}$

Step 7.6

Multiply $- - \frac{3 x^{2} x'}{2 (8 x' + 1)}$ .

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Step 7.6.1

Multiply $- 1$ by $- 1$ .

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} + 1 (\frac{3 x^{2} x'}{2 (8 x' + 1)})$

Step 7.6.2

Multiply $\frac{3 x^{2} x'}{2 (8 x' + 1)}$ by $1$ .

$x^{\frac{1}{2}} = \frac{x'}{2 (8 x' + 1)} + \frac{3 x^{2} x'}{2 (8 x' + 1)}$

Step 8

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

$\frac{d x}{d y} = \frac{x'}{2 (8 x' + 1)} + \frac{3 x^{2} x'}{2 (8 x' + 1)}$