Calculus Examples

$x + y = x y$

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

$1 + y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $g (x) = y$ .

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

Step 3.2

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

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

Step 3.3

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

$x y' + y \cdot 1$

Step 3.4

Multiply $y$ by $1$ .

$x y' + y$

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

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

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

Step 5.2

Subtract $1$ from both sides of the equation.

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

Step 5.3

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

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Step 5.3.1

Factor $y'$ out of ${y'}^{1}$ .

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $1 - x$ .

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Step 5.4.2.1.1

Cancel the common factor.

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

Step 5.4.2.1.2

Divide $y'$ by $1$ .

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

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Combine the numerators over the common denominator.

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

Step 6

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

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