Calculus Examples

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

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 $f (x) = x^{3} + x + 1$ and $g (x) = x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} + x + 1$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [x] + \frac{d}{d x} [1]$ .

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

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $3 x^{2} + 1$ and $0$ .

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

Step 2.6

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

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

Step 2.7

Multiply $- 1$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Simplify the numerator.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

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

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x^{2}$ .

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

Step 3.3.1.2.2

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

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Step 3.3.1.2.2.1

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

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

Step 3.3.1.2.2.2

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

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

Step 3.3.1.2.3

Add $2$ and $1$ .

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

Step 3.3.1.3

Multiply $x$ by $1$ .

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

Step 3.3.1.4

Multiply $- 1$ by $1$ .

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

Step 3.3.2

Combine the opposite terms in $3 x^{3} + x - x^{3} - x - 1$ .

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Step 3.3.2.1

Subtract $x$ from $x$ .

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

Step 3.3.2.2

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

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

Step 3.3.3

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

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