Calculus Examples

$y = \frac{1}{x + 2}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d y} (y) = \frac{d}{d y} (\frac{1}{x + 2})$

Step 2

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

$1$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d y} [\frac{f (y)}{g (y)}]$ is $\frac{g (y) \frac{d}{d y} [f (y)] - f (y) \frac{d}{d y} [g (y)]}{{g (y)}^{2}}$ where $f (y) = 1$ and $g (y) = x + 2$ .

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

Step 3.2

Differentiate.

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Step 3.2.1

Multiply $- 1$ by $1$ .

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

Step 3.2.2

Since $1$ is constant with respect to $y$ , the derivative of $1$ with respect to $y$ is $0$ .

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

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Multiply $x + 2$ by $0$ .

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

Step 3.2.3.2

Subtract $\frac{d}{d y} [x + 2]$ from $0$ .

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

Step 3.2.3.3

Move the negative in front of the fraction.

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

Step 3.2.4

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $x'$ and $0$ .

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

Step 4

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

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

Step 5

Solve for $x'$ .

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Step 5.1

Rewrite the equation as $- \frac{x'}{{(x + 2)}^{2}} = 1$ .

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

Step 5.2

Divide each term in $- \frac{x'}{{(x + 2)}^{2}} = 1$ by $- 1$ and simplify.

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Step 5.2.1

Divide each term in $- \frac{x'}{{(x + 2)}^{2}} = 1$ by $- 1$ .

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

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Dividing two negative values results in a positive value.

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

Step 5.2.2.2

Divide $\frac{x'}{{(x + 2)}^{2}}$ by $1$ .

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

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $1$ by $- 1$ .

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

Step 5.3

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

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

Step 5.4

Simplify the left side.

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Step 5.4.1

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

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Step 5.4.1.1

Cancel the common factor.

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

Step 5.4.1.2

Rewrite the expression.

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

Step 5.5

Simplify $- {(x + 2)}^{2}$ .

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Step 5.5.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

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

Step 5.5.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 5.5.2.1

Apply the distributive property.

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

Step 5.5.2.2

Apply the distributive property.

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

Step 5.5.2.3

Apply the distributive property.

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

Step 5.5.3

Simplify and combine like terms.

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Step 5.5.3.1

Simplify each term.

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Step 5.5.3.1.1

Multiply $x$ by $x$ .

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

Step 5.5.3.1.2

Move $2$ to the left of $x$ .

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

Step 5.5.3.1.3

Multiply $2$ by $2$ .

$x' = - (x^{2} + 2 x + 2 x + 4)$

Step 5.5.3.2

Add $2 x$ and $2 x$ .

$x' = - (x^{2} + 4 x + 4)$

Step 5.5.4

Apply the distributive property.

$x' = - x^{2} - (4 x) - 1 \cdot 4$

Step 5.5.5

Simplify.

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Step 5.5.5.1

Multiply $4$ by $- 1$ .

$x' = - x^{2} - 4 x - 1 \cdot 4$

Step 5.5.5.2

Multiply $- 1$ by $4$ .

$x' = - x^{2} - 4 x - 4$

Step 6

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

$\frac{d x}{d y} = - x^{2} - 4 x - 4$