Calculus Examples

$[(x^{6} + x) (x^{6} - x)]$

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 $f (x) = x^{6} + x$ and $g (x) = x^{6} - x$ .

$(x^{6} + x) \frac{d}{d x} [x^{6} - x] + (x^{6} - x) \frac{d}{d x} [x^{6} + x]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{6} - x$ with respect to $x$ is $\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- x]$ .

$(x^{6} + x) (\frac{d}{d x} [x^{6}] + \frac{d}{d x} [- x]) + (x^{6} - x) \frac{d}{d x} [x^{6} + x]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$(x^{6} + x) (6 x^{5} + \frac{d}{d x} [- x]) + (x^{6} - x) \frac{d}{d x} [x^{6} + x]$

Step 2.3

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

$(x^{6} + x) (6 x^{5} - \frac{d}{d x} [x]) + (x^{6} - x) \frac{d}{d x} [x^{6} + x]$

Step 2.4

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

$(x^{6} + x) (6 x^{5} - 1 \cdot 1) + (x^{6} - x) \frac{d}{d x} [x^{6} + x]$

Step 2.5

Multiply $- 1$ by $1$ .

$(x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) \frac{d}{d x} [x^{6} + x]$

Step 2.6

By the Sum Rule, the derivative of $x^{6} + x$ with respect to $x$ is $\frac{d}{d x} [x^{6}] + \frac{d}{d x} [x]$ .

$(x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) (\frac{d}{d x} [x^{6}] + \frac{d}{d x} [x])$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$(x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) (6 x^{5} + \frac{d}{d x} [x])$

Step 2.8

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

$(x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) (6 x^{5} + 1)$

Step 3

Simplify.

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Step 3.1

Factor $1$ out of $(x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) (6 x^{5} + 1)$ .

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Step 3.1.1

Factor $1$ out of $(x^{6} + x) (6 x^{5} - 1)$ .

$1 ((x^{6} + x) (6 x^{5} - 1)) + (x^{6} - x) (6 x^{5} + 1)$

Step 3.1.2

Factor $1$ out of $(x^{6} - x) (6 x^{5} + 1)$ .

$1 ((x^{6} + x) (6 x^{5} - 1)) + 1 ((x^{6} - x) (6 x^{5} + 1))$

Step 3.1.3

Factor $1$ out of $1 ((x^{6} + x) (6 x^{5} - 1)) + 1 ((x^{6} - x) (6 x^{5} + 1))$ .

$1 ((x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) (6 x^{5} + 1))$

Step 3.2

Multiply $(x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) (6 x^{5} + 1)$ by $1$ .

$(x^{6} + x) (6 x^{5} - 1) + (x^{6} - x) (6 x^{5} + 1)$

Step 3.3

Reorder terms.

$(6 x^{5} - 1) (x^{6} + x) + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4

Simplify each term.

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Step 3.4.1

Expand $(6 x^{5} - 1) (x^{6} + x)$ using the FOIL Method.

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Step 3.4.1.1

Apply the distributive property.

$6 x^{5} (x^{6} + x) - 1 (x^{6} + x) + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.1.2

Apply the distributive property.

$6 x^{5} x^{6} + 6 x^{5} x - 1 (x^{6} + x) + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.1.3

Apply the distributive property.

$6 x^{5} x^{6} + 6 x^{5} x - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

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 $x^{5}$ by $x^{6}$ by adding the exponents.

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Step 3.4.2.1.1.1

Move $x^{6}$ .

$6 (x^{6} x^{5}) + 6 x^{5} x - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.1.2

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

$6 x^{6 + 5} + 6 x^{5} x - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.1.3

Add $6$ and $5$ .

$6 x^{11} + 6 x^{5} x - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.2

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 3.4.2.1.2.1

Move $x$ .

$6 x^{11} + 6 (x \cdot x^{5}) - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.2.2

Multiply $x$ by $x^{5}$ .

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Step 3.4.2.1.2.2.1

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

$6 x^{11} + 6 (x^{1} x^{5}) - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.2.2.2

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

$6 x^{11} + 6 x^{1 + 5} - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.2.3

Add $1$ and $5$ .

$6 x^{11} + 6 x^{6} - 1 x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.3

Rewrite $- 1 x^{6}$ as $- x^{6}$ .

$6 x^{11} + 6 x^{6} - x^{6} - 1 x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.1.4

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

$6 x^{11} + 6 x^{6} - x^{6} - x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.2.2

Subtract $x^{6}$ from $6 x^{6}$ .

$6 x^{11} + 5 x^{6} - x + (6 x^{5} + 1) (x^{6} - x)$

Step 3.4.3

Expand $(6 x^{5} + 1) (x^{6} - x)$ using the FOIL Method.

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Step 3.4.3.1

Apply the distributive property.

$6 x^{11} + 5 x^{6} - x + 6 x^{5} (x^{6} - x) + 1 (x^{6} - x)$

Step 3.4.3.2

Apply the distributive property.

$6 x^{11} + 5 x^{6} - x + 6 x^{5} x^{6} + 6 x^{5} (- x) + 1 (x^{6} - x)$

Step 3.4.3.3

Apply the distributive property.

$6 x^{11} + 5 x^{6} - x + 6 x^{5} x^{6} + 6 x^{5} (- x) + 1 x^{6} + 1 (- x)$

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 $x^{5}$ by $x^{6}$ by adding the exponents.

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Step 3.4.4.1.1.1

Move $x^{6}$ .

$6 x^{11} + 5 x^{6} - x + 6 (x^{6} x^{5}) + 6 x^{5} (- x) + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.1.2

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

$6 x^{11} + 5 x^{6} - x + 6 x^{6 + 5} + 6 x^{5} (- x) + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.1.3

Add $6$ and $5$ .

$6 x^{11} + 5 x^{6} - x + 6 x^{11} + 6 x^{5} (- x) + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.2

Rewrite using the commutative property of multiplication.

$6 x^{11} + 5 x^{6} - x + 6 x^{11} + 6 \cdot - 1 x^{5} x + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.3

Multiply $x^{5}$ by $x$ by adding the exponents.

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Step 3.4.4.1.3.1

Move $x$ .

$6 x^{11} + 5 x^{6} - x + 6 x^{11} + 6 \cdot - 1 (x \cdot x^{5}) + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.3.2

Multiply $x$ by $x^{5}$ .

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Step 3.4.4.1.3.2.1

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

$6 x^{11} + 5 x^{6} - x + 6 x^{11} + 6 \cdot - 1 (x^{1} x^{5}) + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.3.2.2

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

$6 x^{11} + 5 x^{6} - x + 6 x^{11} + 6 \cdot - 1 x^{1 + 5} + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.3.3

Add $1$ and $5$ .

$6 x^{11} + 5 x^{6} - x + 6 x^{11} + 6 \cdot - 1 x^{6} + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.4

Multiply $6$ by $- 1$ .

$6 x^{11} + 5 x^{6} - x + 6 x^{11} - 6 x^{6} + 1 x^{6} + 1 (- x)$

Step 3.4.4.1.5

Multiply $x^{6}$ by $1$ .

$6 x^{11} + 5 x^{6} - x + 6 x^{11} - 6 x^{6} + x^{6} + 1 (- x)$

Step 3.4.4.1.6

Multiply $- x$ by $1$ .

$6 x^{11} + 5 x^{6} - x + 6 x^{11} - 6 x^{6} + x^{6} - x$

Step 3.4.4.2

Add $- 6 x^{6}$ and $x^{6}$ .

$6 x^{11} + 5 x^{6} - x + 6 x^{11} - 5 x^{6} - x$

Step 3.5

Combine the opposite terms in $6 x^{11} + 5 x^{6} - x + 6 x^{11} - 5 x^{6} - x$ .

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Step 3.5.1

Subtract $5 x^{6}$ from $5 x^{6}$ .

$6 x^{11} - x + 6 x^{11} + 0 - x$

Step 3.5.2

Add $6 x^{11} - x + 6 x^{11}$ and $0$ .

$6 x^{11} - x + 6 x^{11} - x$

Step 3.6

Add $6 x^{11}$ and $6 x^{11}$ .

$12 x^{11} - x - x$

Step 3.7

Subtract $x$ from $- x$ .

$12 x^{11} - 2 x$