Calculus Examples

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

Step 1

Simplify.

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

Step 2

Differentiate both sides of the equation.

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

Step 3

Differentiate the left side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that $\frac{d}{d y} [f (y) g (y)]$ is $f (y) \frac{d}{d y} [g (y)] + g (y) \frac{d}{d y} [f (y)]$ where $f (y) = 1 - x$ and $g (y) = y$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

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

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

Step 3.2.2

Multiply $1 - x$ by $1$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Add $0$ and $\frac{d}{d y} [- x]$ .

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

Step 3.2.6

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

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

Step 3.3

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

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

Step 3.4

Reorder terms.

$- y x' - x + 1$

Step 4

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

$2 y$

Step 5

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

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

Step 6

Solve for $x'$ .

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Step 6.1

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

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Step 6.1.1

Add $x$ to both sides of the equation.

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

Step 6.1.2

Subtract $1$ from both sides of the equation.

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

Step 6.2

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

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Step 6.2.1

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

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

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.

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

Step 6.2.2.2

Cancel the common factor of $y$ .

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Step 6.2.2.2.1

Cancel the common factor.

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

Step 6.2.2.2.2

Divide $x'$ by $1$ .

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

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Simplify each term.

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Step 6.2.3.1.1

Cancel the common factor of $y$ .

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Step 6.2.3.1.1.1

Cancel the common factor.

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

Step 6.2.3.1.1.2

Rewrite the expression.

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

Step 6.2.3.1.1.3

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

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

Step 6.2.3.1.2

Multiply $- 1$ by $2$ .

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

Step 6.2.3.1.3

Move the negative in front of the fraction.

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

Step 6.2.3.1.4

Dividing two negative values results in a positive value.

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

Step 7

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

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