Calculus Examples

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

Step 1

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 $g (x) = x^{3} - 2 x$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{3} - 2 x$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{3} - 2 x]$

Step 1.2

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

$2 u \frac{d}{d x} [x^{3} - 2 x]$

Step 1.3

Replace all occurrences of $u$ with $x^{3} - 2 x$ .

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

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{3} - 2 x$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2 x]$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $- 2$ by $1$ .

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Multiply $- 2$ by $2$ .

$(2 x^{3} - 4 x) (3 x^{2} - 2)$

Step 3.3

Expand $(2 x^{3} - 4 x) (3 x^{2} - 2)$ using the FOIL Method.

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Step 3.3.1

Apply the distributive property.

$2 x^{3} (3 x^{2} - 2) - 4 x (3 x^{2} - 2)$

Step 3.3.2

Apply the distributive property.

$2 x^{3} (3 x^{2}) + 2 x^{3} \cdot - 2 - 4 x (3 x^{2} - 2)$

Step 3.3.3

Apply the distributive property.

$2 x^{3} (3 x^{2}) + 2 x^{3} \cdot - 2 - 4 x (3 x^{2}) - 4 x \cdot - 2$

Step 3.4

Simplify and combine like terms.

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Step 3.4.1

Simplify each term.

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Step 3.4.1.1

Rewrite using the commutative property of multiplication.

$2 \cdot 3 x^{3} x^{2} + 2 x^{3} \cdot - 2 - 4 x (3 x^{2}) - 4 x \cdot - 2$

Step 3.4.1.2

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

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Step 3.4.1.2.1

Move $x^{2}$ .

$2 \cdot 3 (x^{2} x^{3}) + 2 x^{3} \cdot - 2 - 4 x (3 x^{2}) - 4 x \cdot - 2$

Step 3.4.1.2.2

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

$2 \cdot 3 x^{2 + 3} + 2 x^{3} \cdot - 2 - 4 x (3 x^{2}) - 4 x \cdot - 2$

Step 3.4.1.2.3

Add $2$ and $3$ .

$2 \cdot 3 x^{5} + 2 x^{3} \cdot - 2 - 4 x (3 x^{2}) - 4 x \cdot - 2$

Step 3.4.1.3

Multiply $2$ by $3$ .

$6 x^{5} + 2 x^{3} \cdot - 2 - 4 x (3 x^{2}) - 4 x \cdot - 2$

Step 3.4.1.4

Multiply $- 2$ by $2$ .

$6 x^{5} - 4 x^{3} - 4 x (3 x^{2}) - 4 x \cdot - 2$

Step 3.4.1.5

Rewrite using the commutative property of multiplication.

$6 x^{5} - 4 x^{3} - 4 \cdot 3 x \cdot x^{2} - 4 x \cdot - 2$

Step 3.4.1.6

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

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Step 3.4.1.6.1

Move $x^{2}$ .

$6 x^{5} - 4 x^{3} - 4 \cdot 3 (x^{2} x) - 4 x \cdot - 2$

Step 3.4.1.6.2

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

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Step 3.4.1.6.2.1

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

$6 x^{5} - 4 x^{3} - 4 \cdot 3 (x^{2} x^{1}) - 4 x \cdot - 2$

Step 3.4.1.6.2.2

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

$6 x^{5} - 4 x^{3} - 4 \cdot 3 x^{2 + 1} - 4 x \cdot - 2$

Step 3.4.1.6.3

Add $2$ and $1$ .

$6 x^{5} - 4 x^{3} - 4 \cdot 3 x^{3} - 4 x \cdot - 2$

Step 3.4.1.7

Multiply $- 4$ by $3$ .

$6 x^{5} - 4 x^{3} - 12 x^{3} - 4 x \cdot - 2$

Step 3.4.1.8

Multiply $- 2$ by $- 4$ .

$6 x^{5} - 4 x^{3} - 12 x^{3} + 8 x$

Step 3.4.2

Subtract $12 x^{3}$ from $- 4 x^{3}$ .

$6 x^{5} - 16 x^{3} + 8 x$

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