Calculus Examples

$[(x^{3} + x) (x^{3} - x)]$

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

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

Step 2

Differentiate.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 2.4

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

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

Step 2.5

Multiply $- 1$ by $1$ .

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

Step 2.6

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

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

Step 2.7

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

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

Step 2.8

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

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

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.2

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

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

Step 3.3

Reorder terms.

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

Step 3.4

Simplify each term.

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Step 3.4.1

Expand $(3 x^{2} - 1) (x^{3} + x)$ using the FOIL Method.

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Step 3.4.1.1

Apply the distributive property.

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

Step 3.4.1.2

Apply the distributive property.

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

Step 3.4.1.3

Apply the distributive property.

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

Step 3.4.2

Simplify and combine like terms.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 3.4.2.1.1.1

Move $x^{3}$ .

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

Step 3.4.2.1.1.2

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

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

Step 3.4.2.1.1.3

Add $3$ and $2$ .

$3 x^{5} + 3 x^{2} x - 1 x^{3} - 1 x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.2.1.2

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

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Step 3.4.2.1.2.1

Move $x$ .

$3 x^{5} + 3 (x \cdot x^{2}) - 1 x^{3} - 1 x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.2.1.2.2

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

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Step 3.4.2.1.2.2.1

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

$3 x^{5} + 3 (x^{1} x^{2}) - 1 x^{3} - 1 x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.2.1.2.2.2

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

$3 x^{5} + 3 x^{1 + 2} - 1 x^{3} - 1 x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.2.1.2.3

Add $1$ and $2$ .

$3 x^{5} + 3 x^{3} - 1 x^{3} - 1 x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.2.1.3

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$3 x^{5} + 3 x^{3} - x^{3} - 1 x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.2.1.4

Rewrite $- 1 x$ as $- x$ .

$3 x^{5} + 3 x^{3} - x^{3} - x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.2.2

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

$3 x^{5} + 2 x^{3} - x + (3 x^{2} + 1) (x^{3} - x)$

Step 3.4.3

Expand $(3 x^{2} + 1) (x^{3} - x)$ using the FOIL Method.

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Step 3.4.3.1

Apply the distributive property.

$3 x^{5} + 2 x^{3} - x + 3 x^{2} (x^{3} - x) + 1 (x^{3} - x)$

Step 3.4.3.2

Apply the distributive property.

$3 x^{5} + 2 x^{3} - x + 3 x^{2} x^{3} + 3 x^{2} (- x) + 1 (x^{3} - x)$

Step 3.4.3.3

Apply the distributive property.

$3 x^{5} + 2 x^{3} - x + 3 x^{2} x^{3} + 3 x^{2} (- x) + 1 x^{3} + 1 (- x)$

Step 3.4.4

Simplify and combine like terms.

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Step 3.4.4.1

Simplify each term.

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Step 3.4.4.1.1

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

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Step 3.4.4.1.1.1

Move $x^{3}$ .

$3 x^{5} + 2 x^{3} - x + 3 (x^{3} x^{2}) + 3 x^{2} (- x) + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.1.2

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

$3 x^{5} + 2 x^{3} - x + 3 x^{3 + 2} + 3 x^{2} (- x) + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.1.3

Add $3$ and $2$ .

$3 x^{5} + 2 x^{3} - x + 3 x^{5} + 3 x^{2} (- x) + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.2

Rewrite using the commutative property of multiplication.

$3 x^{5} + 2 x^{3} - x + 3 x^{5} + 3 \cdot - 1 x^{2} x + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.3

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

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Step 3.4.4.1.3.1

Move $x$ .

$3 x^{5} + 2 x^{3} - x + 3 x^{5} + 3 \cdot - 1 (x \cdot x^{2}) + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.3.2

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

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Step 3.4.4.1.3.2.1

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

$3 x^{5} + 2 x^{3} - x + 3 x^{5} + 3 \cdot - 1 (x^{1} x^{2}) + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.3.2.2

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

$3 x^{5} + 2 x^{3} - x + 3 x^{5} + 3 \cdot - 1 x^{1 + 2} + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.3.3

Add $1$ and $2$ .

$3 x^{5} + 2 x^{3} - x + 3 x^{5} + 3 \cdot - 1 x^{3} + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.4

Multiply $3$ by $- 1$ .

$3 x^{5} + 2 x^{3} - x + 3 x^{5} - 3 x^{3} + 1 x^{3} + 1 (- x)$

Step 3.4.4.1.5

Multiply $x^{3}$ by $1$ .

$3 x^{5} + 2 x^{3} - x + 3 x^{5} - 3 x^{3} + x^{3} + 1 (- x)$

Step 3.4.4.1.6

Multiply $- x$ by $1$ .

$3 x^{5} + 2 x^{3} - x + 3 x^{5} - 3 x^{3} + x^{3} - x$

Step 3.4.4.2

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

$3 x^{5} + 2 x^{3} - x + 3 x^{5} - 2 x^{3} - x$

Step 3.5

Combine the opposite terms in $3 x^{5} + 2 x^{3} - x + 3 x^{5} - 2 x^{3} - x$ .

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Step 3.5.1

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

$3 x^{5} - x + 3 x^{5} + 0 - x$

Step 3.5.2

Add $3 x^{5} - x + 3 x^{5}$ and $0$ .

$3 x^{5} - x + 3 x^{5} - x$

Step 3.6

Add $3 x^{5}$ and $3 x^{5}$ .

$6 x^{5} - x - x$

Step 3.7

Subtract $x$ from $- x$ .

$6 x^{5} - 2 x$