Calculus Examples

$- y + 2 x y = - 3 x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (- y + 2 x y) = \frac{d}{d x} (- 3 x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x y$ with respect to $x$ is $2 \frac{d}{d x} [x y]$ .

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

Step 2.3.2

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 $g (x) = y$ .

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

Step 2.3.3

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

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

Step 2.3.4

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

$- y' + 2 (x y' + y \cdot 1)$

Step 2.3.5

Multiply $y$ by $1$ .

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

Step 2.4

Simplify.

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Step 2.4.1

Apply the distributive property.

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

Step 2.4.2

Remove unnecessary parentheses.

$- y' + 2 x y' + 2 y$

Step 2.4.3

Reorder terms.

$2 x y' + 2 y - y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3 x$ with respect to $x$ is $- 3 \frac{d}{d x} [x]$ .

$- 3 \frac{d}{d x} [x]$

Step 3.2

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

$- 3 \cdot 1$

Step 3.3

Multiply $- 3$ by $1$ .

$- 3$

Step 4

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

$2 x y' + 2 y - y' = - 3$

Step 5

Solve for $y'$ .

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Step 5.1

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

$2 x y' - y' = - 3 - 2 y$

Step 5.2

Factor $y'$ out of $2 x y' - y'$ .

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Step 5.2.1

Factor $y'$ out of $2 x y'$ .

$y' (2 x) - y' = - 3 - 2 y$

Step 5.2.2

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

$y' (2 x) + y' \cdot - 1 = - 3 - 2 y$

Step 5.2.3

Factor $y'$ out of $y' (2 x) + y' \cdot - 1$ .

$y' (2 x - 1) = - 3 - 2 y$

Step 5.3

Divide each term in $y' (2 x - 1) = - 3 - 2 y$ by $2 x - 1$ and simplify.

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Step 5.3.1

Divide each term in $y' (2 x - 1) = - 3 - 2 y$ by $2 x - 1$ .

$\frac{y' (2 x - 1)}{2 x - 1} = \frac{- 3}{2 x - 1} + \frac{- 2 y}{2 x - 1}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2 x - 1$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{y' (2 x - 1)}{2 x - 1} = \frac{- 3}{2 x - 1} + \frac{- 2 y}{2 x - 1}$

Step 5.3.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 3}{2 x - 1} + \frac{- 2 y}{2 x - 1}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Move the negative in front of the fraction.

$y' = - \frac{3}{2 x - 1} + \frac{- 2 y}{2 x - 1}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{3}{2 x - 1} - \frac{2 y}{2 x - 1}$

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 3 - 2 y}{2 x - 1}$

Step 5.3.3.2.2

Rewrite $- 3$ as $- 1 (3)$ .

$y' = \frac{- 1 (3) - 2 y}{2 x - 1}$

Step 5.3.3.2.3

Factor $- 1$ out of $- 2 y$ .

$y' = \frac{- 1 (3) - (2 y)}{2 x - 1}$

Step 5.3.3.2.4

Factor $- 1$ out of $- 1 (3) - (2 y)$ .

$y' = \frac{- 1 (3 + 2 y)}{2 x - 1}$

Step 5.3.3.2.5

Move the negative in front of the fraction.

$y' = - \frac{3 + 2 y}{2 x - 1}$

Step 6

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

$\frac{d y}{d x} = - \frac{3 + 2 y}{2 x - 1}$