Algebra Examples

$y x^{2} + x = 6 + 4 y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y x^{2} + x) = \frac{d}{d x} (6 + 4 y)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Move $2$ to the left of $y$ .

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

Step 2.3

Differentiate using the Power Rule.

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Step 2.3.1

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

$2 y x + x^{2} y' + 1$

Step 2.3.2

Reorder terms.

$2 x y + x^{2} y' + 1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate.

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Step 3.1.1

By the Sum Rule, the derivative of $6 + 4 y$ with respect to $x$ is $\frac{d}{d x} [6] + \frac{d}{d x} [4 y]$ .

$\frac{d}{d x} [6] + \frac{d}{d x} [4 y]$

Step 3.1.2

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

$0 + \frac{d}{d x} [4 y]$

Step 3.2

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

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Step 3.2.1

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

$0 + 4 \frac{d}{d x} [y]$

Step 3.2.2

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

$0 + 4 y'$

Step 3.3

Add $0$ and $4 y'$ .

$4 y'$

Step 4

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

$2 x y + x^{2} y' + 1 = 4 y'$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $4 y'$ from both sides of the equation.

$2 x y + x^{2} y' + 1 - 4 y' = 0$

Step 5.2

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

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Step 5.2.1

Subtract $2 x y$ from both sides of the equation.

$x^{2} y' + 1 - 4 y' = - 2 x y$

Step 5.2.2

Subtract $1$ from both sides of the equation.

$x^{2} y' - 4 y' = - 2 x y - 1$

Step 5.3

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

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Step 5.3.1

Factor $y'$ out of $x^{2} y'$ .

$y' x^{2} - 4 y' = - 2 x y - 1$

Step 5.3.2

Factor $y'$ out of $- 4 y'$ .

$y' x^{2} + y' \cdot - 4 = - 2 x y - 1$

Step 5.3.3

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

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

Step 5.4

Rewrite $4$ as $2^{2}$ .

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

Step 5.5

Factor.

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Step 5.5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$y' ((x + 2) (x - 2)) = - 2 x y - 1$

Step 5.5.2

Remove unnecessary parentheses.

$y' (x + 2) (x - 2) = - 2 x y - 1$

Step 5.6

Divide each term in $y' (x + 2) (x - 2) = - 2 x y - 1$ by $(x + 2) (x - 2)$ and simplify.

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Step 5.6.1

Divide each term in $y' (x + 2) (x - 2) = - 2 x y - 1$ by $(x + 2) (x - 2)$ .

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

Step 5.6.2

Simplify the left side.

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Step 5.6.2.1

Cancel the common factor of $x + 2$ .

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Step 5.6.2.1.1

Cancel the common factor.

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

Step 5.6.2.1.2

Rewrite the expression.

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

Step 5.6.2.2

Cancel the common factor of $x - 2$ .

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Step 5.6.2.2.1

Cancel the common factor.

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

Step 5.6.2.2.2

Divide $y'$ by $1$ .

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

Step 5.6.3

Simplify the right side.

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Step 5.6.3.1

Simplify each term.

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Step 5.6.3.1.1

Move the negative in front of the fraction.

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

Step 5.6.3.1.2

Move the negative in front of the fraction.

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

Step 6

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

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