Calculus Examples

$- 3 = x y - y$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- 3) = \frac{d}{d x} (x y - y)$

Step 2

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

$0$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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 $g (x) = y$ .

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Multiply $y$ by $1$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

$x y' + y - y'$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $x y' + y - y' = 0$ .

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

Step 5.2

Subtract $y$ from both sides of the equation.

$x y' - y' = - y$

Step 5.3

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

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Step 5.3.1

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

$y' x - y' = - y$

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

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.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Move the negative in front of the fraction.

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

Step 6

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

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