Calculus Examples

$6 x {(x^{2} - 1)}^{2}$

Step 1

Since $6$ is constant with respect to $x$ , the derivative of $6 x {(x^{2} - 1)}^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x {(x^{2} - 1)}^{2}]$ .

$6 \frac{d}{d x} [x {(x^{2} - 1)}^{2}]$

Step 2

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 $g (x) = {(x^{2} - 1)}^{2}$ .

$6 (x \frac{d}{d x} [{(x^{2} - 1)}^{2}] + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = x^{2} - 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{2} - 1$ .

$6 (x (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 3.2

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

$6 (x (2 u \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 3.3

Replace all occurrences of $u$ with $x^{2} - 1$ .

$6 (x (2 (x^{2} - 1) \frac{d}{d x} [x^{2} - 1]) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 4

Differentiate.

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Step 4.1

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

$6 (x (2 (x^{2} - 1) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 1])) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 4.2

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

$6 (x (2 (x^{2} - 1) (2 x + \frac{d}{d x} [- 1])) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 4.3

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

$6 (x (2 (x^{2} - 1) (2 x + 0)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $2 x$ and $0$ .

$6 (x (2 (x^{2} - 1) (2 x)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 4.4.2

Multiply $2$ by $2$ .

$6 (x (4 (x^{2} - 1) x) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 5

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

$6 (x^{1} x (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 6

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

$6 (x^{1} x^{1} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 7

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

$6 (x^{1 + 1} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 8

Add $1$ and $1$ .

$6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \frac{d}{d x} [x])$

Step 9

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

$6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2} \cdot 1)$

Step 10

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

$6 (x^{2} (4 (x^{2} - 1)) + {(x^{2} - 1)}^{2})$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$6 (x^{2} (4 x^{2} + 4 \cdot - 1) + {(x^{2} - 1)}^{2})$

Step 11.2

Apply the distributive property.

$6 (x^{2} (4 x^{2}) + x^{2} (4 \cdot - 1) + {(x^{2} - 1)}^{2})$

Step 11.3

Apply the distributive property.

$6 (x^{2} (4 x^{2})) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 11.4

Combine terms.

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Step 11.4.1

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

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Step 11.4.1.1

Move $x^{2}$ .

$6 (x^{2} x^{2} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 11.4.1.2

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

$6 (x^{2 + 2} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 11.4.1.3

Add $2$ and $2$ .

$6 (x^{4} \cdot 4) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 11.4.2

Move $4$ to the left of $x^{4}$ .

$6 (4 \cdot x^{4}) + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 11.4.3

Multiply $4$ by $6$ .

$24 x^{4} + 6 (x^{2} (4 \cdot - 1)) + 6 {(x^{2} - 1)}^{2}$

Step 11.4.4

Multiply $4$ by $- 1$ .

$24 x^{4} + 6 (x^{2} \cdot - 4) + 6 {(x^{2} - 1)}^{2}$

Step 11.4.5

Move $- 4$ to the left of $x^{2}$ .

$24 x^{4} + 6 (- 4 \cdot x^{2}) + 6 {(x^{2} - 1)}^{2}$

Step 11.4.6

Multiply $- 4$ by $6$ .

$24 x^{4} - 24 x^{2} + 6 {(x^{2} - 1)}^{2}$

Step 11.5

Simplify each term.

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Step 11.5.1

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

$24 x^{4} - 24 x^{2} + 6 ((x^{2} - 1) (x^{2} - 1))$

Step 11.5.2

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

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Step 11.5.2.1

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 (x^{2} (x^{2} - 1) - 1 (x^{2} - 1))$

Step 11.5.2.2

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 (x^{2} x^{2} + x^{2} \cdot - 1 - 1 (x^{2} - 1))$

Step 11.5.2.3

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 (x^{2} x^{2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 11.5.3

Simplify and combine like terms.

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Step 11.5.3.1

Simplify each term.

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Step 11.5.3.1.1

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

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Step 11.5.3.1.1.1

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

$24 x^{4} - 24 x^{2} + 6 (x^{2 + 2} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 11.5.3.1.1.2

Add $2$ and $2$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} + x^{2} \cdot - 1 - 1 x^{2} - 1 \cdot - 1)$

Step 11.5.3.1.2

Move $- 1$ to the left of $x^{2}$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} - 1 \cdot x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 11.5.3.1.3

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - 1 x^{2} - 1 \cdot - 1)$

Step 11.5.3.1.4

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - x^{2} - 1 \cdot - 1)$

Step 11.5.3.1.5

Multiply $- 1$ by $- 1$ .

$24 x^{4} - 24 x^{2} + 6 (x^{4} - x^{2} - x^{2} + 1)$

Step 11.5.3.2

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

$24 x^{4} - 24 x^{2} + 6 (x^{4} - 2 x^{2} + 1)$

Step 11.5.4

Apply the distributive property.

$24 x^{4} - 24 x^{2} + 6 x^{4} + 6 (- 2 x^{2}) + 6 \cdot 1$

Step 11.5.5

Simplify.

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Step 11.5.5.1

Multiply $- 2$ by $6$ .

$24 x^{4} - 24 x^{2} + 6 x^{4} - 12 x^{2} + 6 \cdot 1$

Step 11.5.5.2

Multiply $6$ by $1$ .

$24 x^{4} - 24 x^{2} + 6 x^{4} - 12 x^{2} + 6$

Step 11.6

Add $24 x^{4}$ and $6 x^{4}$ .

$30 x^{4} - 24 x^{2} - 12 x^{2} + 6$

Step 11.7

Subtract $12 x^{2}$ from $- 24 x^{2}$ .

$30 x^{4} - 36 x^{2} + 6$