Calculus Examples

$x y - x = y + 2$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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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 $f (x) = x$ and $g (x) = y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.2.4

Multiply $y$ by $1$ .

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

Step 2.3

Evaluate $\frac{d}{d x} [- x]$ .

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Step 2.3.1

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

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

Step 2.3.2

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

$x y' + y - 1 \cdot 1$

Step 2.3.3

Multiply $- 1$ by $1$ .

$x y' + y - 1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$y' + 0$

Step 3.4

Add $y'$ and $0$ .

$y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

$x y' + y - 1 - y' = 0$

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 $y$ from both sides of the equation.

$x y' - 1 - y' = - y$

Step 5.2.2

Add $1$ to both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

Factor $y'$ out of $x y'$ .

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

Step 5.3.2

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

$y' x + y' \cdot - 1 = - y + 1$

Step 5.3.3

Factor $y'$ out of $y' x + y' \cdot - 1$ .

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $x - 1$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

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

Step 5.4.3.2

Factor $- 1$ out of $- y$ .

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

Step 5.4.3.3

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 5.4.3.4

Factor $- 1$ out of $- (y) - 1 (- 1)$ .

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

Step 5.4.3.5

Simplify the expression.

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Step 5.4.3.5.1

Rewrite $- (y - 1)$ as $- 1 (y - 1)$ .

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

Step 5.4.3.5.2

Move the negative in front of the fraction.

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

Step 6

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

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