Calculus Examples

$\frac{2 - \frac{1}{x}}{x - 3}$

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 $f (x) = 2 - \frac{1}{x}$ and $g (x) = x - 3$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $2 - \frac{1}{x}$ with respect to $x$ is $\frac{d}{d x} [2] + \frac{d}{d x} [- \frac{1}{x}]$ .

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

Step 2.2

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

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

Step 2.3

Add $0$ and $\frac{d}{d x} [- \frac{1}{x}]$ .

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

Step 3

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 $f (x) = - 1$ and $g (x) = \frac{1}{x}$ .

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

Step 4

Differentiate.

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Step 4.1

Rewrite $\frac{1}{x}$ as $x^{- 1}$ .

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

Step 4.2

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

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

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

Multiply $x^{- 2}$ by $1$ .

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

Step 4.4

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

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

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $\frac{1}{x}$ by $0$ .

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

Step 4.5.2

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 4.8

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

$\frac{(x - 3) x^{- 2} - (2 - \frac{1}{x}) (1 + 0)}{{(x - 3)}^{2}}$

Step 4.9

Simplify the expression.

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Step 4.9.1

Add $1$ and $0$ .

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

Step 4.9.2

Multiply $- 1$ by $1$ .

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

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Simplify each term.

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Step 5.4.1.1

Cancel the common factor of $x$ .

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Step 5.4.1.1.1

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

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

Step 5.4.1.1.2

Cancel the common factor.

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

Step 5.4.1.1.3

Rewrite the expression.

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

Step 5.4.1.2

Combine $- 3$ and $\frac{1}{x^{2}}$ .

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

Step 5.4.1.3

Move the negative in front of the fraction.

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

Step 5.4.1.4

Multiply $- 1$ by $2$ .

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

Step 5.4.1.5

Multiply $- - \frac{1}{x}$ .

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Step 5.4.1.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.4.1.5.2

Multiply $\frac{1}{x}$ by $1$ .

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

Step 5.4.2

Combine the numerators over the common denominator.

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

Step 5.4.3

Add $1$ and $1$ .

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

Step 5.4.4

Move the negative in front of the fraction.

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

Step 5.5

Combine terms.

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Step 5.5.1

Multiply $\frac{- 2 + \frac{2}{x} - \frac{3}{x^{2}}}{{(x - 3)}^{2}}$ by $\frac{x}{x}$ .

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

Step 5.5.2

Combine.

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

Step 5.5.3

Apply the distributive property.

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

Step 5.5.4

Cancel the common factor of $x$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

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

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

Step 5.5.6

Combine $x$ and $\frac{3}{x^{2}}$ .

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

Step 5.5.7

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

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

Step 5.5.8

Cancel the common factor of $x$ and $x^{2}$ .

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Step 5.5.8.1

Factor $x$ out of $3 x$ .

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

Step 5.5.8.2

Cancel the common factors.

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Step 5.5.8.2.1

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

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

Step 5.5.8.2.2

Cancel the common factor.

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

Step 5.5.8.2.3

Rewrite the expression.

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

Step 5.6

Reorder terms.

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

Step 5.7

Simplify the numerator.

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Step 5.7.1

To write $- 2 x$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.7.2

Combine $- 2 x$ and $\frac{x}{x}$ .

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

Step 5.7.3

Combine the numerators over the common denominator.

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

Step 5.7.4

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

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Step 5.7.4.1

Move $x$ .

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

Step 5.7.4.2

Multiply $x$ by $x$ .

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

Step 5.7.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 5.7.6

Combine the numerators over the common denominator.

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

Step 5.7.7

Reorder terms.

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

Step 5.8

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.9

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

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Step 5.9.1

Multiply $\frac{- 2 x^{2} + 2 x - 3}{x}$ by $\frac{1}{x {(x - 3)}^{2}}$ .

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

Step 5.9.2

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

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

Step 5.9.3

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

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

Step 5.9.4

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

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

Step 5.9.5

Add $1$ and $1$ .

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

Step 5.10

Factor $- 1$ out of $- 2 x^{2}$ .

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

Step 5.11

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

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

Step 5.12

Factor $- 1$ out of $- (2 x^{2}) - (- 2 x)$ .

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

Step 5.13

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

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

Step 5.14

Factor $- 1$ out of $- (2 x^{2} - 2 x) - 1 (3)$ .

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

Step 5.15

Rewrite $- (2 x^{2} - 2 x + 3)$ as $- 1 (2 x^{2} - 2 x + 3)$ .

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

Step 5.16

Move the negative in front of the fraction.

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