Calculus Examples

$\frac{x^{2} - 2 x + 2}{x - 1}$

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 $g (x) = x - 1$ .

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

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $- 2$ by $1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

By the Sum Rule, the derivative of $x - 1$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 1]$ .

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

Step 2.9

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

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

Step 2.10

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

$\frac{(x - 1) (2 x - 2) - (x^{2} - 2 x + 2) (1 + 0)}{{(x - 1)}^{2}}$

Step 2.11

Simplify the expression.

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Step 2.11.1

Add $1$ and $0$ .

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

Step 2.11.2

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

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 $(x - 1) (2 x - 2)$ using the FOIL Method.

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Step 3.2.1.1.1

Apply the distributive property.

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

Step 3.2.1.1.2

Apply the distributive property.

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

Step 3.2.1.1.3

Apply the distributive property.

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

Step 3.2.1.2

Simplify and combine like terms.

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Step 3.2.1.2.1

Simplify each term.

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Step 3.2.1.2.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.2.1.2.1.2

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

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Step 3.2.1.2.1.2.1

Move $x$ .

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

Step 3.2.1.2.1.2.2

Multiply $x$ by $x$ .

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

Step 3.2.1.2.1.3

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

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

Step 3.2.1.2.1.4

Multiply $2$ by $- 1$ .

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

Step 3.2.1.2.1.5

Multiply $- 1$ by $- 2$ .

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

Step 3.2.1.2.2

Subtract $2 x$ from $- 2 x$ .

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

Step 3.2.1.3

Multiply $- 2$ by $- 1$ .

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

Step 3.2.1.4

Multiply $- 1$ by $2$ .

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

Step 3.2.2

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

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Step 3.2.2.1

Subtract $2$ from $2$ .

$\frac{2 x^{2} - 4 x - x^{2} + 2 x + 0}{{(x - 1)}^{2}}$

Step 3.2.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Add $- 4 x$ and $2 x$ .

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

Step 3.3

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

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Factor $x$ out of $x \cdot x + x \cdot - 2$ .

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