Calculus Examples

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

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $\frac{x}{3}$ and ${(2 x + 1)}^{3}$ .

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

Step 1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x {(2 x + 1)}^{3}}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x {(2 x + 1)}^{3}]$ .

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

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 $g (x) = {(2 x + 1)}^{3}$ .

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

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 2 x + 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $2 x + 1$ .

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

Step 3.2

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

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

Step 3.3

Replace all occurrences of $u$ with $2 x + 1$ .

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

Step 4

Differentiate.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Multiply $2$ by $1$ .

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

Step 4.5

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

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

Step 4.6

Simplify the expression.

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Step 4.6.1

Add $2$ and $0$ .

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

Step 4.6.2

Multiply $2$ by $3$ .

$\frac{1}{3} (x (6 {(2 x + 1)}^{2}) + {(2 x + 1)}^{3} \frac{d}{d x} [x])$

Step 4.7

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

$\frac{1}{3} (x (6 {(2 x + 1)}^{2}) + {(2 x + 1)}^{3} \cdot 1)$

Step 4.8

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

$\frac{1}{3} (x (6 {(2 x + 1)}^{2}) + {(2 x + 1)}^{3})$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$\frac{1}{3} (x (6 {(2 x + 1)}^{2})) + \frac{1}{3} {(2 x + 1)}^{3}$

Step 5.2

Combine $x$ and $\frac{1}{3}$ .

$\frac{x}{3} (6 {(2 x + 1)}^{2}) + \frac{1}{3} {(2 x + 1)}^{3}$

Step 5.3

Combine $6$ and $\frac{x}{3}$ .

$\frac{6 x}{3} {(2 x + 1)}^{2} + \frac{1}{3} {(2 x + 1)}^{3}$

Step 5.4

Combine $\frac{6 x}{3}$ and ${(2 x + 1)}^{2}$ .

$\frac{6 x {(2 x + 1)}^{2}}{3} + \frac{1}{3} {(2 x + 1)}^{3}$

Step 5.5

Cancel the common factor of $6$ and $3$ .

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Step 5.5.1

Factor $3$ out of $6 x {(2 x + 1)}^{2}$ .

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

Step 5.5.2

Cancel the common factors.

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Step 5.5.2.1

Factor $3$ out of $3$ .

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

Step 5.5.2.2

Cancel the common factor.

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

Step 5.5.2.3

Rewrite the expression.

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

Step 5.5.2.4

Divide $2 x {(2 x + 1)}^{2}$ by $1$ .

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

Step 5.6

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

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Step 5.6.1

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

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

Step 5.6.2

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

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

Step 5.6.3

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

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

Step 5.7

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

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

Step 5.8

Expand $(2 x + 1) (2 x + 1)$ using the FOIL Method.

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Step 5.8.1

Apply the distributive property.

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

Step 5.8.2

Apply the distributive property.

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

Step 5.8.3

Apply the distributive property.

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

Step 5.9

Simplify and combine like terms.

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Step 5.9.1

Simplify each term.

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Step 5.9.1.1

Rewrite using the commutative property of multiplication.

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

Step 5.9.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.9.1.2.1

Move $x$ .

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

Step 5.9.1.2.2

Multiply $x$ by $x$ .

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

Step 5.9.1.3

Multiply $2$ by $2$ .

$(4 x^{2} + 2 x \cdot 1 + 1 (2 x) + 1 \cdot 1) (2 x + \frac{1}{3} (2 x + 1))$

Step 5.9.1.4

Multiply $2$ by $1$ .

$(4 x^{2} + 2 x + 1 (2 x) + 1 \cdot 1) (2 x + \frac{1}{3} (2 x + 1))$

Step 5.9.1.5

Multiply $2 x$ by $1$ .

$(4 x^{2} + 2 x + 2 x + 1 \cdot 1) (2 x + \frac{1}{3} (2 x + 1))$

Step 5.9.1.6

Multiply $1$ by $1$ .

$(4 x^{2} + 2 x + 2 x + 1) (2 x + \frac{1}{3} (2 x + 1))$

Step 5.9.2

Add $2 x$ and $2 x$ .

$(4 x^{2} + 4 x + 1) (2 x + \frac{1}{3} (2 x + 1))$

Step 5.10

Simplify each term.

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Step 5.10.1

Apply the distributive property.

$(4 x^{2} + 4 x + 1) (2 x + \frac{1}{3} (2 x) + \frac{1}{3} \cdot 1)$

Step 5.10.2

Multiply $\frac{1}{3} (2 x)$ .

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Step 5.10.2.1

Combine $2$ and $\frac{1}{3}$ .

$(4 x^{2} + 4 x + 1) (2 x + \frac{2}{3} x + \frac{1}{3} \cdot 1)$

Step 5.10.2.2

Combine $\frac{2}{3}$ and $x$ .

$(4 x^{2} + 4 x + 1) (2 x + \frac{2 x}{3} + \frac{1}{3} \cdot 1)$

Step 5.10.3

Multiply $\frac{1}{3}$ by $1$ .

$(4 x^{2} + 4 x + 1) (2 x + \frac{2 x}{3} + \frac{1}{3})$

Step 5.11

To write $2 x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$(4 x^{2} + 4 x + 1) (2 x \cdot \frac{3}{3} + \frac{2 x}{3} + \frac{1}{3})$

Step 5.12

Combine $2 x$ and $\frac{3}{3}$ .

$(4 x^{2} + 4 x + 1) (\frac{2 x \cdot 3}{3} + \frac{2 x}{3} + \frac{1}{3})$

Step 5.13

Combine the numerators over the common denominator.

$(4 x^{2} + 4 x + 1) (\frac{2 x \cdot 3 + 2 x}{3} + \frac{1}{3})$

Step 5.14

Combine the numerators over the common denominator.

$(4 x^{2} + 4 x + 1) \frac{2 x \cdot 3 + 2 x + 1}{3}$

Step 5.15

Multiply $3$ by $2$ .

$(4 x^{2} + 4 x + 1) \frac{6 x + 2 x + 1}{3}$

Step 5.16

Add $6 x$ and $2 x$ .

$(4 x^{2} + 4 x + 1) \frac{8 x + 1}{3}$

Step 5.17

Apply the distributive property.

$4 x^{2} \frac{8 x + 1}{3} + 4 x \frac{8 x + 1}{3} + 1 \frac{8 x + 1}{3}$

Step 5.18

Simplify.

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Step 5.18.1

Multiply $4 x^{2} \frac{8 x + 1}{3}$ .

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Step 5.18.1.1

Combine $4$ and $\frac{8 x + 1}{3}$ .

$x^{2} \frac{4 (8 x + 1)}{3} + 4 x \frac{8 x + 1}{3} + 1 \frac{8 x + 1}{3}$

Step 5.18.1.2

Combine $x^{2}$ and $\frac{4 (8 x + 1)}{3}$ .

$\frac{x^{2} (4 (8 x + 1))}{3} + 4 x \frac{8 x + 1}{3} + 1 \frac{8 x + 1}{3}$

Step 5.18.2

Multiply $4 x \frac{8 x + 1}{3}$ .

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Step 5.18.2.1

Combine $4$ and $\frac{8 x + 1}{3}$ .

$\frac{x^{2} (4 (8 x + 1))}{3} + x \frac{4 (8 x + 1)}{3} + 1 \frac{8 x + 1}{3}$

Step 5.18.2.2

Combine $x$ and $\frac{4 (8 x + 1)}{3}$ .

$\frac{x^{2} (4 (8 x + 1))}{3} + \frac{x (4 (8 x + 1))}{3} + 1 \frac{8 x + 1}{3}$

Step 5.18.3

Multiply $\frac{8 x + 1}{3}$ by $1$ .

$\frac{x^{2} (4 (8 x + 1))}{3} + \frac{x (4 (8 x + 1))}{3} + \frac{8 x + 1}{3}$

Step 5.19

Combine the numerators over the common denominator.

$\frac{x^{2} (4 (8 x + 1)) + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20

Simplify each term.

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Step 5.20.1

Rewrite using the commutative property of multiplication.

$\frac{4 x^{2} (8 x + 1) + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.2

Apply the distributive property.

$\frac{4 x^{2} (8 x) + 4 x^{2} \cdot 1 + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.3

Rewrite using the commutative property of multiplication.

$\frac{4 \cdot 8 x^{2} x + 4 x^{2} \cdot 1 + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.4

Multiply $4$ by $1$ .

$\frac{4 \cdot 8 x^{2} x + 4 x^{2} + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.5

Simplify each term.

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Step 5.20.5.1

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

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Step 5.20.5.1.1

Move $x$ .

$\frac{4 \cdot 8 (x \cdot x^{2}) + 4 x^{2} + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.5.1.2

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

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Step 5.20.5.1.2.1

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

$\frac{4 \cdot 8 (x^{1} x^{2}) + 4 x^{2} + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.5.1.2.2

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

$\frac{4 \cdot 8 x^{1 + 2} + 4 x^{2} + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.5.1.3

Add $1$ and $2$ .

$\frac{4 \cdot 8 x^{3} + 4 x^{2} + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.5.2

Multiply $4$ by $8$ .

$\frac{32 x^{3} + 4 x^{2} + x (4 (8 x + 1)) + 8 x + 1}{3}$

Step 5.20.6

Rewrite using the commutative property of multiplication.

$\frac{32 x^{3} + 4 x^{2} + 4 x (8 x + 1) + 8 x + 1}{3}$

Step 5.20.7

Apply the distributive property.

$\frac{32 x^{3} + 4 x^{2} + 4 x (8 x) + 4 x \cdot 1 + 8 x + 1}{3}$

Step 5.20.8

Rewrite using the commutative property of multiplication.

$\frac{32 x^{3} + 4 x^{2} + 4 \cdot 8 x \cdot x + 4 x \cdot 1 + 8 x + 1}{3}$

Step 5.20.9

Multiply $4$ by $1$ .

$\frac{32 x^{3} + 4 x^{2} + 4 \cdot 8 x \cdot x + 4 x + 8 x + 1}{3}$

Step 5.20.10

Simplify each term.

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Step 5.20.10.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.20.10.1.1

Move $x$ .

$\frac{32 x^{3} + 4 x^{2} + 4 \cdot 8 (x \cdot x) + 4 x + 8 x + 1}{3}$

Step 5.20.10.1.2

Multiply $x$ by $x$ .

$\frac{32 x^{3} + 4 x^{2} + 4 \cdot 8 x^{2} + 4 x + 8 x + 1}{3}$

Step 5.20.10.2

Multiply $4$ by $8$ .

$\frac{32 x^{3} + 4 x^{2} + 32 x^{2} + 4 x + 8 x + 1}{3}$

Step 5.21

Add $4 x^{2}$ and $32 x^{2}$ .

$\frac{32 x^{3} + 36 x^{2} + 4 x + 8 x + 1}{3}$

Step 5.22

Add $4 x$ and $8 x$ .

$\frac{32 x^{3} + 36 x^{2} + 12 x + 1}{3}$