Algebra Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

$2 (2 y)$

Step 2.3

Multiply $2$ by $2$ .

$4 y$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (y) = 5 x - 3$ and $g (y) = 5 x + 3$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $5 x - 3$ with respect to $y$ is $\frac{d}{d y} [5 x] + \frac{d}{d y} [- 3]$ .

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

Step 3.2.2

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

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

Step 3.3

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

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

Step 3.4

Since $- 3$ is constant with respect to $y$ , the derivative of $- 3$ with respect to $y$ is $0$ .

$\frac{(5 x + 3) (5 x' + 0) - (5 x - 3) \frac{d}{d y} [5 x + 3]}{{(5 x + 3)}^{2}}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $5 x'$ and $0$ .

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

Step 3.5.2

Move $5$ to the left of $5 x + 3$ .

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

Step 3.6

By the Sum Rule, the derivative of $5 x + 3$ with respect to $y$ is $\frac{d}{d y} [5 x] + \frac{d}{d y} [3]$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

Since $3$ is constant with respect to $y$ , the derivative of $3$ with respect to $y$ is $0$ .

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

Step 3.10

Simplify the expression.

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Step 3.10.1

Add $5 x'$ and $0$ .

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

Step 3.10.2

Multiply $5$ by $- 1$ .

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

Step 3.11

Simplify.

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Step 3.11.1

Apply the distributive property.

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

Step 3.11.2

Apply the distributive property.

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

Step 3.11.3

Apply the distributive property.

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

Step 3.11.4

Apply the distributive property.

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

Step 3.11.5

Simplify the numerator.

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Step 3.11.5.1

Combine the opposite terms in $5 (5 x) x' + 5 \cdot 3 x' - 5 (5 x) x' - 5 \cdot - 3 x'$ .

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Step 3.11.5.1.1

Subtract $5 (5 x) x'$ from $5 (5 x) x'$ .

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

Step 3.11.5.1.2

Add $5 \cdot 3 x'$ and $0$ .

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

Step 3.11.5.2

Simplify each term.

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Step 3.11.5.2.1

Multiply $5$ by $3$ .

$\frac{15 x' - 5 \cdot - 3 x'}{{(5 x + 3)}^{2}}$

Step 3.11.5.2.2

Multiply $- 5$ by $- 3$ .

$\frac{15 x' + 15 x'}{{(5 x + 3)}^{2}}$

Step 3.11.5.3

Add $15 x'$ and $15 x'$ .

$\frac{30 x'}{{(5 x + 3)}^{2}}$

Step 4

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

$4 y = \frac{30 x'}{{(5 x + 3)}^{2}}$

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $\frac{30 x'}{{(5 x + 3)}^{2}} = 4 y$ .

$\frac{30 x'}{{(5 x + 3)}^{2}} = 4 y$

Step 5.2

Multiply both sides by ${(5 x + 3)}^{2}$ .

$\frac{30 x'}{{(5 x + 3)}^{2}} {(5 x + 3)}^{2} = 4 y {(5 x + 3)}^{2}$

Step 5.3

Simplify the left side.

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Step 5.3.1

Cancel the common factor of ${(5 x + 3)}^{2}$ .

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Step 5.3.1.1

Cancel the common factor.

$\frac{30 x'}{{(5 x + 3)}^{2}} {(5 x + 3)}^{2} = 4 y {(5 x + 3)}^{2}$

Step 5.3.1.2

Rewrite the expression.

$30 x' = 4 y {(5 x + 3)}^{2}$

Step 5.4

Divide each term in $30 x' = 4 y {(5 x + 3)}^{2}$ by $30$ and simplify.

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Step 5.4.1

Divide each term in $30 x' = 4 y {(5 x + 3)}^{2}$ by $30$ .

$\frac{30 x'}{30} = \frac{4 y {(5 x + 3)}^{2}}{30}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $30$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{30 x'}{30} = \frac{4 y {(5 x + 3)}^{2}}{30}$

Step 5.4.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{4 y {(5 x + 3)}^{2}}{30}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Cancel the common factor of $4$ and $30$ .

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Step 5.4.3.1.1

Factor $2$ out of $4 y {(5 x + 3)}^{2}$ .

$x' = \frac{2 (2 y {(5 x + 3)}^{2})}{30}$

Step 5.4.3.1.2

Cancel the common factors.

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Step 5.4.3.1.2.1

Factor $2$ out of $30$ .

$x' = \frac{2 (2 y {(5 x + 3)}^{2})}{2 (15)}$

Step 5.4.3.1.2.2

Cancel the common factor.

$x' = \frac{2 (2 y {(5 x + 3)}^{2})}{2 \cdot 15}$

Step 5.4.3.1.2.3

Rewrite the expression.

$x' = \frac{2 y {(5 x + 3)}^{2}}{15}$

Step 6

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

$\frac{d x}{d y} = \frac{2 y {(5 x + 3)}^{2}}{15}$