Calculus Examples

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

Step 1

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

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

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Multiply.

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

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

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

Step 4.4

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

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

Step 4.5

Simplify the expression.

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Step 4.5.1

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

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

Step 4.5.2

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

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

Step 4.6

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

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

Step 4.7

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

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

Step 5

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

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Add $1$ and $1$ .

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Apply the distributive property.

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

Step 11.3

Combine terms.

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Step 11.3.1

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

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Step 11.3.1.1

Move $x$ .

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

Step 11.3.1.2

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

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Step 11.3.1.2.1

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

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

Step 11.3.1.2.2

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

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

Step 11.3.1.3

Add $1$ and $2$ .

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

Step 11.3.2

Move $2$ to the left of $x^{3}$ .

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

Step 11.3.3

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

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

Step 11.3.4

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

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Step 11.3.4.1

Cancel the common factor.

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

Step 11.3.4.2

Rewrite the expression.

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

Step 11.3.5

Subtract $1$ from $1$ .

$x^{2} + \frac{2 x^{3} + 0}{x}$

Step 11.3.6

Add $2 x^{3}$ and $0$ .

$x^{2} + \frac{2 x^{3}}{x}$

Step 11.3.7

Cancel the common factor of $x^{3}$ and $x$ .

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Step 11.3.7.1

Factor $x$ out of $2 x^{3}$ .

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

Step 11.3.7.2

Cancel the common factors.

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Step 11.3.7.2.1

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

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

Step 11.3.7.2.2

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

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

Step 11.3.7.2.3

Cancel the common factor.

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

Step 11.3.7.2.4

Rewrite the expression.

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

Step 11.3.7.2.5

Divide $2 x^{2}$ by $1$ .

$x^{2} + 2 x^{2}$

Step 11.3.8

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

$3 x^{2}$