Calculus Examples

$\frac{2}{\sqrt{x}} + \frac{3}{\sqrt{y}} = 6$

Step 1

Rewrite the left side with rational exponents.

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Step 1.1

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

$\frac{2}{x^{\frac{1}{2}}} + \frac{3}{\sqrt{y}} = 6$

Step 1.2

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

$\frac{2}{x^{\frac{1}{2}}} + \frac{3}{y^{\frac{1}{2}}} = 6$

Step 2

Differentiate both sides of the equation.

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

Step 3

Differentiate the left side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{\frac{1}{2}}$ .

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

Step 3.2.3.2

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

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

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $x^{\frac{1}{2}}$ .

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

Step 3.2.4

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

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

Step 3.2.5

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

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Step 3.2.5.1

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

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

Step 3.2.5.2

Cancel the common factor of $2$ .

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Step 3.2.5.2.1

Factor $2$ out of $- 2$ .

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

Step 3.2.5.2.2

Cancel the common factor.

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

Step 3.2.5.2.3

Rewrite the expression.

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

Combine the numerators over the common denominator.

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

Step 3.2.9

Simplify the numerator.

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Step 3.2.9.1

Multiply $- 1$ by $2$ .

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

Step 3.2.9.2

Subtract $2$ from $1$ .

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

Step 3.2.10

Move the negative in front of the fraction.

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

Step 3.2.11

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

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

Step 3.2.12

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

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

Step 3.2.13

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

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Step 3.2.13.1

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

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

Step 3.2.13.2

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

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

Step 3.2.13.3

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

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

Step 3.2.13.4

Combine the numerators over the common denominator.

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

Step 3.2.13.5

Simplify the numerator.

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Step 3.2.13.5.1

Multiply $- 1$ by $2$ .

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

Step 3.2.13.5.2

Subtract $2$ from $- 1$ .

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

Step 3.2.13.6

Move the negative in front of the fraction.

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

Step 3.2.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2.15

Multiply $- 1$ by $2$ .

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

Step 3.2.16

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

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

Step 3.2.17

Factor $2$ out of $- 2$ .

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

Step 3.2.18

Cancel the common factors.

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Step 3.2.18.1

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

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

Step 3.2.18.2

Cancel the common factor.

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

Step 3.2.18.3

Rewrite the expression.

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

Step 3.2.19

Move the negative in front of the fraction.

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

Step 3.3

Evaluate $\frac{d}{d x} [\frac{3}{y^{\frac{1}{2}}}]$ .

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Step 3.3.1

Since $3$ is constant with respect to $x$ , the derivative of $\frac{3}{y^{\frac{1}{2}}}$ with respect to $x$ is $3 \frac{d}{d x} [\frac{1}{y^{\frac{1}{2}}}]$ .

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

Step 3.3.2

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 $f (x) = 1$ and $g (x) = y^{\frac{1}{2}}$ .

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

Step 3.3.3

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

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

Step 3.3.4

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

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Step 3.3.4.1

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

$- \frac{1}{x^{\frac{3}{2}}} + 3 \frac{y^{\frac{1}{2}} \cdot 0 - 1 \cdot 1 (\frac{d}{d u_{2}} [{u_{2}}^{\frac{1}{2}}] \frac{d}{d x} [y])}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.4.2

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

$- \frac{1}{x^{\frac{3}{2}}} + 3 \frac{y^{\frac{1}{2}} \cdot 0 - 1 \cdot 1 (\frac{1}{2} {u_{2}}^{\frac{1}{2} - 1} \frac{d}{d x} [y])}{{(y^{\frac{1}{2}})}^{2}}$

Step 3.3.4.3

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

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

Step 3.3.5

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

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

Step 3.3.6

Multiply $y^{\frac{1}{2}}$ by $0$ .

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

Step 3.3.7

Multiply $- 1$ by $1$ .

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

Step 3.3.8

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

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

Step 3.3.9

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

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

Step 3.3.10

Combine the numerators over the common denominator.

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

Step 3.3.11

Simplify the numerator.

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Step 3.3.11.1

Multiply $- 1$ by $2$ .

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

Step 3.3.11.2

Subtract $2$ from $1$ .

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

Step 3.3.12

Move the negative in front of the fraction.

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

Step 3.3.13

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

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

Step 3.3.14

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

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

Step 3.3.15

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

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

Step 3.3.16

Subtract $\frac{y'}{2 y^{\frac{1}{2}}}$ from $0$ .

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

Step 3.3.17

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

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Step 3.3.17.1

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

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

Step 3.3.17.2

Cancel the common factor of $2$ .

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Step 3.3.17.2.1

Cancel the common factor.

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

Step 3.3.17.2.2

Rewrite the expression.

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

Step 3.3.18

Simplify.

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

Step 3.3.19

Rewrite $\frac{- \frac{y'}{2 y^{\frac{1}{2}}}}{y}$ as a product.

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

Step 3.3.20

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

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

Step 3.3.21

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

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

Step 3.3.22

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

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

Step 3.3.23

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

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

Step 3.3.24

Combine the numerators over the common denominator.

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

Step 3.3.25

Add $2$ and $1$ .

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

Step 3.3.26

Multiply $- 1$ by $3$ .

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

Step 3.3.27

Combine $- 3$ and $\frac{y'}{2 y^{\frac{3}{2}}}$ .

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

Step 3.3.28

Move the negative in front of the fraction.

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

Step 4

Since $6$ is constant with respect to $x$ , the derivative of $6$ with respect to $x$ is $0$ .

$0$

Step 5

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

$- \frac{1}{x^{\frac{3}{2}}} - \frac{3 y'}{2 y^{\frac{3}{2}}} = 0$

Step 6

Solve for $y'$ .

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Step 6.1

Add $\frac{1}{x^{\frac{3}{2}}}$ to both sides of the equation.

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

Step 6.2

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

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Step 6.2.1

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

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

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.2.2.2

Divide $\frac{3 y'}{2 y^{\frac{3}{2}}}$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Move the negative one from the denominator of $\frac{\frac{1}{x^{\frac{3}{2}}}}{- 1}$ .

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

Step 6.2.3.2

Rewrite $- 1 \cdot \frac{1}{x^{\frac{3}{2}}}$ as $- \frac{1}{x^{\frac{3}{2}}}$ .

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

Step 6.3

Multiply both sides by $2 y^{\frac{3}{2}}$ .

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

Step 6.4

Simplify.

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Step 6.4.1

Simplify the left side.

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Step 6.4.1.1

Simplify $\frac{3 y'}{2 y^{\frac{3}{2}}} (2 y^{\frac{3}{2}})$ .

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Step 6.4.1.1.1

Rewrite using the commutative property of multiplication.

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

Step 6.4.1.1.2

Cancel the common factor of $2$ .

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Step 6.4.1.1.2.1

Cancel the common factor.

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

Step 6.4.1.1.2.2

Rewrite the expression.

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

Step 6.4.1.1.3

Cancel the common factor of $y^{\frac{3}{2}}$ .

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Step 6.4.1.1.3.1

Cancel the common factor.

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

Step 6.4.1.1.3.2

Rewrite the expression.

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

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

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

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Step 6.4.2.1.1

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

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Step 6.4.2.1.1.1

Multiply $2$ by $- 1$ .

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

Step 6.4.2.1.1.2

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

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

Step 6.4.2.1.1.3

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

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

Step 6.4.2.1.2

Move the negative in front of the fraction.

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

Step 6.5

Divide each term in $3 y' = - \frac{2 y^{\frac{3}{2}}}{x^{\frac{3}{2}}}$ by $3$ and simplify.

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Step 6.5.1

Divide each term in $3 y' = - \frac{2 y^{\frac{3}{2}}}{x^{\frac{3}{2}}}$ by $3$ .

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

Step 6.5.2

Simplify the left side.

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Step 6.5.2.1

Cancel the common factor of $3$ .

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Step 6.5.2.1.1

Cancel the common factor.

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

Step 6.5.2.1.2

Divide $y'$ by $1$ .

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

Step 6.5.3

Simplify the right side.

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Step 6.5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.5.3.2

Multiply $\frac{1}{3}$ by $\frac{2 y^{\frac{3}{2}}}{x^{\frac{3}{2}}}$ .

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

Step 7

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

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