Calculus Examples

$\sqrt{y} = 2 x^{4} y + \frac{5}{x^{2}}$

Step 1

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

$y^{\frac{1}{2}} = 2 x^{4} y + \frac{5}{x^{2}}$

Step 2

Differentiate both sides of the equation.

$\frac{d}{d x} (y^{\frac{1}{2}}) = \frac{d}{d x} (2 x^{4} y + \frac{5}{x^{2}})$

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

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Step 3.1.1

To apply the Chain Rule, set $u$ as $y$ .

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

Step 3.1.2

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

$\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [y]$

Step 3.1.3

Replace all occurrences of $u$ with $y$ .

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

Step 3.2

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

$\frac{1}{2} y^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [y]$

Step 3.3

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

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

Step 3.4

Combine the numerators over the common denominator.

$\frac{1}{2} y^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [y]$

Step 3.5

Simplify the numerator.

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Step 3.5.1

Multiply $- 1$ by $2$ .

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

Step 3.5.2

Subtract $2$ from $1$ .

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

Step 3.6

Move the negative in front of the fraction.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

$\frac{1}{2 y^{\frac{1}{2}}} y'$

Step 3.10

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

$\frac{y'}{2 y^{\frac{1}{2}}}$

Step 4

Differentiate the right side of the equation.

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Step 4.1

By the Sum Rule, the derivative of $2 x^{4} y + \frac{5}{x^{2}}$ with respect to $x$ is $\frac{d}{d x} [2 x^{4} y] + \frac{d}{d x} [\frac{5}{x^{2}}]$ .

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

Step 4.2

Evaluate $\frac{d}{d x} [2 x^{4} y]$ .

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Step 4.2.1

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

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

Step 4.2.2

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 $f (x) = x^{4}$ and $g (x) = y$ .

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

Step 4.2.3

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

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

Step 4.2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

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

Step 4.2.5

Move $4$ to the left of $y$ .

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

Step 4.3

Evaluate $\frac{d}{d x} [\frac{5}{x^{2}}]$ .

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Step 4.3.1

Since $5$ is constant with respect to $x$ , the derivative of $\frac{5}{x^{2}}$ with respect to $x$ is $5 \frac{d}{d x} [\frac{1}{x^{2}}]$ .

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

Step 4.3.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

Step 4.3.3

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

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Step 4.3.3.1

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

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

Step 4.3.3.2

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

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

Step 4.3.3.3

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

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

Step 4.3.4

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

$2 (x^{4} y' + 4 y x^{3}) + 5 (- {(x^{2})}^{- 2} (2 x))$

Step 4.3.5

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

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Step 4.3.5.1

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

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

Step 4.3.5.2

Multiply $2$ by $- 2$ .

$2 (x^{4} y' + 4 y x^{3}) + 5 (- x^{- 4} (2 x))$

Step 4.3.6

Multiply $2$ by $- 1$ .

$2 (x^{4} y' + 4 y x^{3}) + 5 (- 2 x^{- 4} x)$

Step 4.3.7

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

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

Step 4.3.8

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

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

Step 4.3.9

Subtract $4$ from $1$ .

$2 (x^{4} y' + 4 y x^{3}) + 5 (- 2 x^{- 3})$

Step 4.3.10

Multiply $- 2$ by $5$ .

$2 (x^{4} y' + 4 y x^{3}) - 10 x^{- 3}$

Step 4.4

Simplify.

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Step 4.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (x^{4} y' + 4 y x^{3}) - 10 \frac{1}{x^{3}}$

Step 4.4.2

Apply the distributive property.

$2 (x^{4} y') + 2 (4 y x^{3}) - 10 \frac{1}{x^{3}}$

Step 4.4.3

Combine terms.

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Step 4.4.3.1

Multiply $4$ by $2$ .

$2 x^{4} y' + 8 (y x^{3}) - 10 \frac{1}{x^{3}}$

Step 4.4.3.2

Combine $- 10$ and $\frac{1}{x^{3}}$ .

$2 x^{4} y' + 8 y x^{3} + \frac{- 10}{x^{3}}$

Step 4.4.3.3

Move the negative in front of the fraction.

$2 x^{4} y' + 8 y x^{3} - \frac{10}{x^{3}}$

Step 4.4.4

Reorder terms.

$2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}$

Step 5

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

$\frac{y'}{2 y^{\frac{1}{2}}} = 2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}$

Step 6

Solve for $y'$ .

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Step 6.1

Multiply both sides by $2 y^{\frac{1}{2}}$ .

$\frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{1}{2}}) = (2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}) (2 y^{\frac{1}{2}})$

Step 6.2

Simplify.

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Step 6.2.1

Simplify the left side.

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Step 6.2.1.1

Simplify $\frac{y'}{2 y^{\frac{1}{2}}} (2 y^{\frac{1}{2}})$ .

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Step 6.2.1.1.1

Rewrite using the commutative property of multiplication.

$2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{1}{2}} = (2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}) (2 y^{\frac{1}{2}})$

Step 6.2.1.1.2

Cancel the common factor of $2$ .

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Step 6.2.1.1.2.1

Cancel the common factor.

$2 \frac{y'}{2 y^{\frac{1}{2}}} y^{\frac{1}{2}} = (2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}) (2 y^{\frac{1}{2}})$

Step 6.2.1.1.2.2

Rewrite the expression.

$\frac{y'}{y^{\frac{1}{2}}} y^{\frac{1}{2}} = (2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}) (2 y^{\frac{1}{2}})$

Step 6.2.1.1.3

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

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Step 6.2.1.1.3.1

Cancel the common factor.

$\frac{y'}{y^{\frac{1}{2}}} y^{\frac{1}{2}} = (2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}) (2 y^{\frac{1}{2}})$

Step 6.2.1.1.3.2

Rewrite the expression.

$y' = (2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}) (2 y^{\frac{1}{2}})$

Step 6.2.2

Simplify the right side.

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Step 6.2.2.1

Simplify $(2 x^{4} y' + 8 x^{3} y - \frac{10}{x^{3}}) (2 y^{\frac{1}{2}})$ .

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Step 6.2.2.1.1

Apply the distributive property.

$y' = 2 x^{4} y' (2 y^{\frac{1}{2}}) + 8 x^{3} y (2 y^{\frac{1}{2}}) - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2

Simplify.

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Step 6.2.2.1.2.1

Multiply $2$ by $2$ .

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y (2 y^{\frac{1}{2}}) - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2.2

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

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Step 6.2.2.1.2.2.1

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

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} (y^{\frac{1}{2}} y) \cdot 2 - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2.2.2

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

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Step 6.2.2.1.2.2.2.1

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

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} (y^{\frac{1}{2}} y^{1}) \cdot 2 - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2.2.2.2

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

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{1}{2} + 1} \cdot 2 - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2.2.3

Write $1$ as a fraction with a common denominator.

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{1}{2} + \frac{2}{2}} \cdot 2 - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2.2.4

Combine the numerators over the common denominator.

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{1 + 2}{2}} \cdot 2 - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2.2.5

Add $1$ and $2$ .

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{3}{2}} \cdot 2 - \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$

Step 6.2.2.1.2.3

Multiply $- \frac{10}{x^{3}} (2 y^{\frac{1}{2}})$ .

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Step 6.2.2.1.2.3.1

Multiply $2$ by $- 1$ .

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{3}{2}} \cdot 2 - 2 \frac{10}{x^{3}} y^{\frac{1}{2}}$

Step 6.2.2.1.2.3.2

Combine $- 2$ and $\frac{10}{x^{3}}$ .

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{3}{2}} \cdot 2 + \frac{- 2 \cdot 10}{x^{3}} y^{\frac{1}{2}}$

Step 6.2.2.1.2.3.3

Multiply $- 2$ by $10$ .

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{3}{2}} \cdot 2 + \frac{- 20}{x^{3}} y^{\frac{1}{2}}$

Step 6.2.2.1.2.3.4

Combine $\frac{- 20}{x^{3}}$ and $y^{\frac{1}{2}}$ .

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 8 x^{3} y^{\frac{3}{2}} \cdot 2 + \frac{- 20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.2.2.1.3

Simplify each term.

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Step 6.2.2.1.3.1

Multiply $2$ by $8$ .

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 16 x^{3} y^{\frac{3}{2}} + \frac{- 20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.2.2.1.3.2

Move the negative in front of the fraction.

$y' = 4 x^{4} y' y^{\frac{1}{2}} + 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.2.2.1.4

Move $x^{4}$ .

$y' = 4 y' x^{4} y^{\frac{1}{2}} + 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.3

Solve for $y'$ .

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Step 6.3.1

Subtract $4 y' x^{4} y^{\frac{1}{2}}$ from both sides of the equation.

$y' - 4 y' x^{4} y^{\frac{1}{2}} = 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.3.2

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

${({y'}^{\frac{1}{2}})}^{2} - 4 {(y' x y^{\frac{1}{2}})}^{9} - 16 {(x y^{\frac{1}{2}})}^{9} + \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.3.3

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

${({y'}^{\frac{1}{2}})}^{2} - 4 {(y' x u)}^{9} - 16 {(x (u))}^{9} + \frac{20 (u)}{x^{3}} = 0$

Step 6.3.4

Substitute $y'$ for $u$ .

${({y'}^{\frac{1}{2}})}^{2} - 4 {(y' x y^{\frac{1}{2}})}^{9} - 16 {(x (y^{\frac{1}{2}}))}^{9} + \frac{20 (y^{\frac{1}{2}})}{x^{3}} = 0$

Step 6.3.5

Factor $y'$ out of $y' - 4 y' x^{4} y^{\frac{1}{2}}$ .

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Step 6.3.5.1

Raise $y'$ to the power of $1$ .

$y' - 4 y' x^{4} y^{\frac{1}{2}} = 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.3.5.2

Factor $y'$ out of ${y'}^{1}$ .

$y' \cdot 1 - 4 y' x^{4} y^{\frac{1}{2}} = 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.3.5.3

Factor $y'$ out of $- 4 y' x^{4} y^{\frac{1}{2}}$ .

$y' \cdot 1 + y' (- 4 x^{4} y^{\frac{1}{2}}) = 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.3.5.4

Factor $y'$ out of $y' \cdot 1 + y' (- 4 x^{4} y^{\frac{1}{2}})$ .

$y' (1 - 4 x^{4} y^{\frac{1}{2}}) = 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$

Step 6.3.6

Divide each term in $y' (1 - 4 x^{4} y^{\frac{1}{2}}) = 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$ by $1 - 4 x^{4} y^{\frac{1}{2}}$ and simplify.

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Step 6.3.6.1

Divide each term in $y' (1 - 4 x^{4} y^{\frac{1}{2}}) = 16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}$ by $1 - 4 x^{4} y^{\frac{1}{2}}$ .

$\frac{y' (1 - 4 x^{4} y^{\frac{1}{2}})}{1 - 4 x^{4} y^{\frac{1}{2}}} = \frac{16 x^{3} y^{\frac{3}{2}}}{1 - 4 x^{4} y^{\frac{1}{2}}} + \frac{- \frac{20 y^{\frac{1}{2}}}{x^{3}}}{1 - 4 x^{4} y^{\frac{1}{2}}}$

Step 6.3.6.2

Simplify the left side.

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Step 6.3.6.2.1

Cancel the common factor.

Step 6.3.6.2.2

Divide $y'$ by $1$ .

$y' = \frac{16 x^{3} y^{\frac{3}{2}}}{1 - 4 x^{4} y^{\frac{1}{2}}} + \frac{- \frac{20 y^{\frac{1}{2}}}{x^{3}}}{1 - 4 x^{4} y^{\frac{1}{2}}}$

Step 6.3.6.3

Simplify the right side.

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Step 6.3.6.3.1

Combine the numerators over the common denominator.

$y' = \frac{16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}}{1 - 4 x^{4} y^{\frac{1}{2}}}$

Step 6.3.6.3.2

Multiply the numerator and denominator of the fraction by $x^{3}$ .

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Step 6.3.6.3.2.1

Multiply $\frac{16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}}{1 - 4 x^{4} y^{\frac{1}{2}}}$ by $\frac{x^{3}}{x^{3}}$ .

$y' = \frac{x^{3}}{x^{3}} \cdot \frac{16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}}}{1 - 4 x^{4} y^{\frac{1}{2}}}$

Step 6.3.6.3.2.2

Combine.

$y' = \frac{x^{3} (16 x^{3} y^{\frac{3}{2}} - \frac{20 y^{\frac{1}{2}}}{x^{3}})}{x^{3} (1 - 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.3

Apply the distributive property.

$y' = \frac{x^{3} (16 x^{3} y^{\frac{3}{2}}) + x^{3} (- \frac{20 y^{\frac{1}{2}}}{x^{3}})}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.4

Cancel the common factor of $x^{3}$ .

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Step 6.3.6.3.4.1

Move the leading negative in $- \frac{20 y^{\frac{1}{2}}}{x^{3}}$ into the numerator.

$y' = \frac{x^{3} (16 x^{3} y^{\frac{3}{2}}) + x^{3} \frac{- 20 y^{\frac{1}{2}}}{x^{3}}}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.4.2

Cancel the common factor.

$y' = \frac{x^{3} (16 x^{3} y^{\frac{3}{2}}) + x^{3} \frac{- 20 y^{\frac{1}{2}}}{x^{3}}}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.4.3

Rewrite the expression.

$y' = \frac{x^{3} (16 x^{3} y^{\frac{3}{2}}) - 20 y^{\frac{1}{2}}}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.5

Simplify the numerator.

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Step 6.3.6.3.5.1

Factor $4 y^{\frac{1}{2}}$ out of $x^{3} (16 x^{3} y^{\frac{3}{2}}) - 20 y^{\frac{1}{2}}$ .

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Step 6.3.6.3.5.1.1

Reorder the expression.

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Step 6.3.6.3.5.1.1.1

Move parentheses.

$y' = \frac{x^{3} (16 x^{3}) y^{\frac{3}{2}} - 20 y^{\frac{1}{2}}}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.5.1.1.2

Reorder $x^{3}$ and $16$ .

$y' = \frac{16 \cdot x^{3} x^{3} y^{\frac{3}{2}} - 20 y^{\frac{1}{2}}}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.5.1.2

Factor $4 y^{\frac{1}{2}}$ out of $16 \cdot x^{3} x^{3} y^{\frac{3}{2}}$ .

$y' = \frac{4 y^{\frac{1}{2}} (4 \cdot x^{3} x^{3} y^{\frac{2}{2}}) - 20 y^{\frac{1}{2}}}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.5.1.3

Factor $4 y^{\frac{1}{2}}$ out of $- 20 y^{\frac{1}{2}}$ .

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

Step 6.3.6.3.5.1.4

Factor $4 y^{\frac{1}{2}}$ out of $4 y^{\frac{1}{2}} (4 \cdot x^{3} x^{3} y^{\frac{2}{2}}) + 4 y^{\frac{1}{2}} (- 5)$ .

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

Step 6.3.6.3.5.2

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

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Step 6.3.6.3.5.2.1

Move $x^{3}$ .

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

Step 6.3.6.3.5.2.2

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

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

Step 6.3.6.3.5.2.3

Add $3$ and $3$ .

$y' = \frac{4 y^{\frac{1}{2}} (4 \cdot x^{6} y^{\frac{2}{2}} - 5)}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.5.3

Simplify each term.

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Step 6.3.6.3.5.3.1

Divide $2$ by $2$ .

$y' = \frac{4 y^{\frac{1}{2}} (4 x^{6} y^{1} - 5)}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.5.3.2

Simplify.

$y' = \frac{4 y^{\frac{1}{2}} (4 x^{6} y - 5)}{x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.6

Factor $x^{3}$ out of $x^{3} \cdot 1 + x^{3} (- 4 x^{4} y^{\frac{1}{2}})$ .

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Step 6.3.6.3.6.1

Factor $x^{3}$ out of $x^{3} \cdot 1$ .

$y' = \frac{4 y^{\frac{1}{2}} (4 x^{6} y - 5)}{x^{3} (1) + x^{3} (- 4 x^{4} y^{\frac{1}{2}})}$

Step 6.3.6.3.6.2

Factor $x^{3}$ out of $x^{3} (1) + x^{3} (- 4 x^{4} y^{\frac{1}{2}})$ .

$y' = \frac{4 y^{\frac{1}{2}} (4 x^{6} y - 5)}{x^{3} (1 - 4 x^{4} y^{\frac{1}{2}})}$

Step 7

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

$\frac{d y}{d x} = \frac{4 y^{\frac{1}{2}} (4 x^{6} y - 5)}{x^{3} (1 - 4 x^{4} y^{\frac{1}{2}})}$