Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = x - 2$ and $g (x) = y + 3$ .

$\frac{(y + 3) \frac{d}{d x} [x - 2] - (x - 2) \frac{d}{d x} [y + 3]}{{(y + 3)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

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

$\frac{(y + 3) (\frac{d}{d x} [x] + \frac{d}{d x} [- 2]) - (x - 2) \frac{d}{d x} [y + 3]}{{(y + 3)}^{2}}$

Step 3.2.2

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

$\frac{(y + 3) (1 + \frac{d}{d x} [- 2]) - (x - 2) \frac{d}{d x} [y + 3]}{{(y + 3)}^{2}}$

Step 3.2.3

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

$\frac{(y + 3) (1 + 0) - (x - 2) \frac{d}{d x} [y + 3]}{{(y + 3)}^{2}}$

Step 3.2.4

Simplify the expression.

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Step 3.2.4.1

Add $1$ and $0$ .

$\frac{(y + 3) \cdot 1 - (x - 2) \frac{d}{d x} [y + 3]}{{(y + 3)}^{2}}$

Step 3.2.4.2

Multiply $y + 3$ by $1$ .

$\frac{y + 3 - (x - 2) \frac{d}{d x} [y + 3]}{{(y + 3)}^{2}}$

Step 3.2.5

By the Sum Rule, the derivative of $y + 3$ with respect to $x$ is $\frac{d}{d x} [y] + \frac{d}{d x} [3]$ .

$\frac{y + 3 - (x - 2) (\frac{d}{d x} [y] + \frac{d}{d x} [3])}{{(y + 3)}^{2}}$

Step 3.3

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

$\frac{y + 3 - (x - 2) (y' + \frac{d}{d x} [3])}{{(y + 3)}^{2}}$

Step 3.4

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

$\frac{y + 3 - (x - 2) (y' + 0)}{{(y + 3)}^{2}}$

Step 3.5

Add $y'$ and $0$ .

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

Step 3.6

Simplify.

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Step 3.6.1

Apply the distributive property.

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

Step 3.6.2

Apply the distributive property.

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

Step 3.6.3

Multiply $- 1$ by $- 2$ .

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

Step 3.6.4

Reorder terms.

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

Step 4

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

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

Step 5

Solve for $y'$ .

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Step 5.1

Multiply both sides by ${(y + 3)}^{2}$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{- x y' + y + 2 y' + 3}{{(y + 3)}^{2}} {(y + 3)}^{2}$ .

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Step 5.2.1.1

Cancel the common factor of ${(y + 3)}^{2}$ .

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Step 5.2.1.1.1

Cancel the common factor.

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

Step 5.2.1.1.2

Rewrite the expression.

$y' {(y + 3)}^{2} = - x y' + y + 2 y' + 3$

Step 5.2.1.2

Reorder.

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Step 5.2.1.2.1

Move $x$ .

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

Step 5.2.1.2.2

Move $y$ .

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

Step 5.3

Solve for $y'$ .

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Step 5.3.1

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 5.3.2

Rewrite $- 1 y'$ as $- y'$ .

$- y' x + 2 y' + y + 3 = y' {(y + 3)}^{2}$

Step 5.3.3

Simplify $y' {(y + 3)}^{2}$ .

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Step 5.3.3.1

Rewrite ${(y + 3)}^{2}$ as $(y + 3) (y + 3)$ .

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

Step 5.3.3.2

Expand $(y + 3) (y + 3)$ using the FOIL Method.

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Step 5.3.3.2.1

Apply the distributive property.

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

Step 5.3.3.2.2

Apply the distributive property.

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

Step 5.3.3.2.3

Apply the distributive property.

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

Step 5.3.3.3

Simplify and combine like terms.

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Step 5.3.3.3.1

Simplify each term.

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Step 5.3.3.3.1.1

Multiply $y$ by $y$ .

$- y' x + 2 y' + y + 3 = y' (y^{2} + y \cdot 3 + 3 y + 3 \cdot 3)$

Step 5.3.3.3.1.2

Move $3$ to the left of $y$ .

$- y' x + 2 y' + y + 3 = y' (y^{2} + 3 \cdot y + 3 y + 3 \cdot 3)$

Step 5.3.3.3.1.3

Multiply $3$ by $3$ .

$- y' x + 2 y' + y + 3 = y' (y^{2} + 3 y + 3 y + 9)$

Step 5.3.3.3.2

Add $3 y$ and $3 y$ .

$- y' x + 2 y' + y + 3 = y' (y^{2} + 6 y + 9)$

Step 5.3.3.4

Apply the distributive property.

$- y' x + 2 y' + y + 3 = y' (y^{2}) + y' (6 y) + y' \cdot 9$

Step 5.3.3.5

Simplify.

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Step 5.3.3.5.1

Rewrite using the commutative property of multiplication.

$- y' x + 2 y' + y + 3 = y' y^{2} + 6 y' y + y' \cdot 9$

Step 5.3.3.5.2

Move $9$ to the left of $y'$ .

$- y' x + 2 y' + y + 3 = y' y^{2} + 6 y' y + 9 \cdot y'$

$- y' x + 2 y' + y + 3 = y' y^{2} + 6 y' y + 9 y'$

Step 5.3.4

Since $y'$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$y' y^{2} + 6 y' y + 9 y' = - y' x + 2 y' + y + 3$

Step 5.3.5

Move all terms containing $y'$ to the left side of the equation.

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Step 5.3.5.1

Add $y' x$ to both sides of the equation.

$y' y^{2} + 6 y' y + 9 y' + y' x = 2 y' + y + 3$

Step 5.3.5.2

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

$y' y^{2} + 6 y' y + 9 y' + y' x - 2 y' = y + 3$

Step 5.3.5.3

Subtract $2 y'$ from $9 y'$ .

$y' y^{2} + 6 y' y + y' x + 7 y' = y + 3$

Step 5.3.6

Factor $y'$ out of $y' y^{2} + 6 y' y + y' x + 7 y'$ .

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Step 5.3.6.1

Factor $y'$ out of $y' y^{2}$ .

$y' (y^{2}) + 6 y' y + y' (x) + 7 y' = y + 3$

Step 5.3.6.2

Factor $y'$ out of $6 y' y$ .

$y' (y^{2}) + y' (6 y) + y' (x) + 7 y' = y + 3$

Step 5.3.6.3

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

$y' (y^{2}) + y' (6 y) + y' (x) + 7 y' = y + 3$

Step 5.3.6.4

Factor $y'$ out of $7 y'$ .

$y' (y^{2}) + y' (6 y) + y' (x) + y' \cdot 7 = y + 3$

Step 5.3.6.5

Factor $y'$ out of $y' (y^{2}) + y' (6 y)$ .

$y' (y^{2} + 6 y) + y' (x) + y' \cdot 7 = y + 3$

Step 5.3.6.6

Factor $y'$ out of $y' (y^{2} + 6 y) + y' (x)$ .

$y' (y^{2} + 6 y + x) + y' \cdot 7 = y + 3$

Step 5.3.6.7

Factor $y'$ out of $y' (y^{2} + 6 y + x) + y' \cdot 7$ .

$y' (y^{2} + 6 y + x + 7) = y + 3$

Step 5.3.7

Divide each term in $y' (y^{2} + 6 y + x + 7) = y + 3$ by $y^{2} + 6 y + x + 7$ and simplify.

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Step 5.3.7.1

Divide each term in $y' (y^{2} + 6 y + x + 7) = y + 3$ by $y^{2} + 6 y + x + 7$ .

$\frac{y' (y^{2} + 6 y + x + 7)}{y^{2} + 6 y + x + 7} = \frac{y}{y^{2} + 6 y + x + 7} + \frac{3}{y^{2} + 6 y + x + 7}$

Step 5.3.7.2

Simplify the left side.

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Step 5.3.7.2.1

Cancel the common factor of $y^{2} + 6 y + x + 7$ .

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Step 5.3.7.2.1.1

Cancel the common factor.

$\frac{y' (y^{2} + 6 y + x + 7)}{y^{2} + 6 y + x + 7} = \frac{y}{y^{2} + 6 y + x + 7} + \frac{3}{y^{2} + 6 y + x + 7}$

Step 5.3.7.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{y}{y^{2} + 6 y + x + 7} + \frac{3}{y^{2} + 6 y + x + 7}$

Step 5.3.7.3

Simplify the right side.

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Step 5.3.7.3.1

Combine the numerators over the common denominator.

$y' = \frac{y + 3}{y^{2} + 6 y + x + 7}$

Step 6

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

$\frac{d y}{d x} = \frac{y + 3}{y^{2} + 6 y + x + 7}$