Calculus Examples

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

Step 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 $f (x) = x^{2} + 2 x - 2$ and $g (x) = x^{2} - 2 x + 2$ .

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $2$ by $1$ .

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

Step 2.6

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

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

Step 2.7

Add $2 x + 2$ and $0$ .

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

Step 2.8

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

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

Step 2.9

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.10

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

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

Step 2.11

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

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

Step 2.12

Multiply $- 2$ by $1$ .

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

Step 2.13

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

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

Step 2.14

Add $2 x - 2$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Simplify the numerator.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Expand $(x^{2} - 2 x + 2) (2 x + 2)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.2.1.2

Combine the opposite terms in $x^{2} (2 x) + x^{2} \cdot 2 - 2 x (2 x) - 2 x \cdot 2 + 2 (2 x) + 2 \cdot 2$ .

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Step 3.2.1.2.1

Reorder the factors in the terms $- 2 x \cdot 2$ and $2 (2 x)$ .

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

Step 3.2.1.2.2

Add $- 2 \cdot 2 x$ and $2 \cdot 2 x$ .

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

Step 3.2.1.2.3

Add $x^{2} (2 x) + x^{2} \cdot 2 - 2 x (2 x)$ and $0$ .

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

Step 3.2.1.3

Simplify each term.

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Step 3.2.1.3.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.3.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.2.1.3.2.1

Move $x$ .

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

Step 3.2.1.3.2.2

Multiply $x$ by $x^{2}$ .

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Step 3.2.1.3.2.2.1

Raise $x$ to the power of $1$ .

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

Step 3.2.1.3.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.2.1.3.2.3

Add $1$ and $2$ .

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

Step 3.2.1.3.3

Move $2$ to the left of $x^{2}$ .

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

Step 3.2.1.3.4

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.3.5

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.3.5.1

Move $x$ .

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

Step 3.2.1.3.5.2

Multiply $x$ by $x$ .

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

Step 3.2.1.3.6

Multiply $- 2$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} - 4 x^{2} + 2 \cdot 2 + (- x^{2} - (2 x) - - 2) (2 x - 2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.3.7

Multiply $2$ by $2$ .

$\frac{2 x^{3} + 2 x^{2} - 4 x^{2} + 4 + (- x^{2} - (2 x) - - 2) (2 x - 2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.4

Subtract $4 x^{2}$ from $2 x^{2}$ .

$\frac{2 x^{3} - 2 x^{2} + 4 + (- x^{2} - (2 x) - - 2) (2 x - 2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.5

Simplify each term.

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Step 3.2.1.5.1

Multiply $2$ by $- 1$ .

$\frac{2 x^{3} - 2 x^{2} + 4 + (- x^{2} - 2 x - - 2) (2 x - 2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.5.2

Multiply $- 1$ by $- 2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 + (- x^{2} - 2 x + 2) (2 x - 2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.6

Expand $(- x^{2} - 2 x + 2) (2 x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{2 x^{3} - 2 x^{2} + 4 - x^{2} (2 x) - x^{2} \cdot - 2 - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7

Simplify each term.

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Step 3.2.1.7.1

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 2 x^{2} + 4 - 1 \cdot 2 x^{2} x - x^{2} \cdot - 2 - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.2.1.7.2.1

Move $x$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 1 \cdot 2 (x \cdot x^{2}) - x^{2} \cdot - 2 - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.2.2

Multiply $x$ by $x^{2}$ .

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Step 3.2.1.7.2.2.1

Raise $x$ to the power of $1$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 1 \cdot 2 (x^{1} x^{2}) - x^{2} \cdot - 2 - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 x^{3} - 2 x^{2} + 4 - 1 \cdot 2 x^{1 + 2} - x^{2} \cdot - 2 - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.2.3

Add $1$ and $2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 1 \cdot 2 x^{3} - x^{2} \cdot - 2 - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.3

Multiply $- 1$ by $2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} - x^{2} \cdot - 2 - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.4

Multiply $- 2$ by $- 1$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 2 x (2 x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.5

Rewrite using the commutative property of multiplication.

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 2 \cdot 2 x \cdot x - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.6

Multiply $x$ by $x$ by adding the exponents.

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Step 3.2.1.7.6.1

Move $x$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 2 \cdot 2 (x \cdot x) - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.6.2

Multiply $x$ by $x$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 2 \cdot 2 x^{2} - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.7

Multiply $- 2$ by $2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 4 x^{2} - 2 x \cdot - 2 + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.8

Multiply $- 2$ by $- 2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 4 x^{2} + 4 x + 2 (2 x) + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.9

Multiply $2$ by $2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 4 x^{2} + 4 x + 4 x + 2 \cdot - 2}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.7.10

Multiply $2$ by $- 2$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} + 2 x^{2} - 4 x^{2} + 4 x + 4 x - 4}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.8

Subtract $4 x^{2}$ from $2 x^{2}$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} - 2 x^{2} + 4 x + 4 x - 4}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.1.9

Add $4 x$ and $4 x$ .

$\frac{2 x^{3} - 2 x^{2} + 4 - 2 x^{3} - 2 x^{2} + 8 x - 4}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.2

Combine the opposite terms in $2 x^{3} - 2 x^{2} + 4 - 2 x^{3} - 2 x^{2} + 8 x - 4$ .

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Step 3.2.2.1

Subtract $2 x^{3}$ from $2 x^{3}$ .

$\frac{- 2 x^{2} + 4 + 0 - 2 x^{2} + 8 x - 4}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.2.2

Add $- 2 x^{2} + 4$ and $0$ .

$\frac{- 2 x^{2} + 4 - 2 x^{2} + 8 x - 4}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.2.3

Subtract $4$ from $4$ .

$\frac{- 2 x^{2} - 2 x^{2} + 8 x + 0}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.2.4

Add $- 2 x^{2} - 2 x^{2} + 8 x$ and $0$ .

$\frac{- 2 x^{2} - 2 x^{2} + 8 x}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.2.3

Subtract $2 x^{2}$ from $- 2 x^{2}$ .

$\frac{- 4 x^{2} + 8 x}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.3

Factor $4 x$ out of $- 4 x^{2} + 8 x$ .

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Step 3.3.1

Factor $4 x$ out of $- 4 x^{2}$ .

$\frac{4 x (- x) + 8 x}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.3.2

Factor $4 x$ out of $8 x$ .

$\frac{4 x (- x) + 4 x (2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.3.3

Factor $4 x$ out of $4 x (- x) + 4 x (2)$ .

$\frac{4 x (- x + 2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.4

Factor $- 1$ out of $- x$ .

$\frac{4 x (- (x) + 2)}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.5

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{4 x (- (x) - 1 (- 2))}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.6

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

$\frac{4 x (- (x - 2))}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.7

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

$\frac{4 x (- 1 (x - 2))}{{(x^{2} - 2 x + 2)}^{2}}$

Step 3.8

Move the negative in front of the fraction.

$- \frac{(4 x) (x - 2)}{{(x^{2} - 2 x + 2)}^{2}}$