Calculus Examples

$f (x) = {(2 x + 7)}^{3}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{3}$ and $g (x) = 2 x + 7$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $2 x + 7$ .

$\frac{d}{d u} [u^{3}] \frac{d}{d x} [2 x + 7]$

Step 1.1.2

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

$3 u^{2} \frac{d}{d x} [2 x + 7]$

Step 1.1.3

Replace all occurrences of $u$ with $2 x + 7$ .

$3 {(2 x + 7)}^{2} \frac{d}{d x} [2 x + 7]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $2 x + 7$ with respect to $x$ is $\frac{d}{d x} [2 x] + \frac{d}{d x} [7]$ .

$3 {(2 x + 7)}^{2} (\frac{d}{d x} [2 x] + \frac{d}{d x} [7])$

Step 1.2.2

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

$3 {(2 x + 7)}^{2} (2 \frac{d}{d x} [x] + \frac{d}{d x} [7])$

Step 1.2.3

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

$3 {(2 x + 7)}^{2} (2 \cdot 1 + \frac{d}{d x} [7])$

Step 1.2.4

Multiply $2$ by $1$ .

$3 {(2 x + 7)}^{2} (2 + \frac{d}{d x} [7])$

Step 1.2.5

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$3 {(2 x + 7)}^{2} (2 + 0)$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $2$ and $0$ .

$3 {(2 x + 7)}^{2} \cdot 2$

Step 1.2.6.2

Multiply $2$ by $3$ .

$f' (x) = 6 {(2 x + 7)}^{2}$

Step 2

Find the second derivative.

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Step 2.1

Rewrite ${(2 x + 7)}^{2}$ as $(2 x + 7) (2 x + 7)$ .

$\frac{d}{d x} [6 ((2 x + 7) (2 x + 7))]$

Step 2.2

Expand $(2 x + 7) (2 x + 7)$ using the FOIL Method.

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Step 2.2.1

Apply the distributive property.

$\frac{d}{d x} [6 (2 x (2 x + 7) + 7 (2 x + 7))]$

Step 2.2.2

Apply the distributive property.

$\frac{d}{d x} [6 (2 x (2 x) + 2 x \cdot 7 + 7 (2 x + 7))]$

Step 2.2.3

Apply the distributive property.

$\frac{d}{d x} [6 (2 x (2 x) + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7)]$

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [6 (2 \cdot 2 x \cdot x + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7)]$

Step 2.3.1.2

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

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Step 2.3.1.2.1

Move $x$ .

$\frac{d}{d x} [6 (2 \cdot 2 (x \cdot x) + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7)]$

Step 2.3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [6 (2 \cdot 2 x^{2} + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7)]$

Step 2.3.1.3

Multiply $2$ by $2$ .

$\frac{d}{d x} [6 (4 x^{2} + 2 x \cdot 7 + 7 (2 x) + 7 \cdot 7)]$

Step 2.3.1.4

Multiply $7$ by $2$ .

$\frac{d}{d x} [6 (4 x^{2} + 14 x + 7 (2 x) + 7 \cdot 7)]$

Step 2.3.1.5

Multiply $2$ by $7$ .

$\frac{d}{d x} [6 (4 x^{2} + 14 x + 14 x + 7 \cdot 7)]$

Step 2.3.1.6

Multiply $7$ by $7$ .

$\frac{d}{d x} [6 (4 x^{2} + 14 x + 14 x + 49)]$

Step 2.3.2

Add $14 x$ and $14 x$ .

$\frac{d}{d x} [6 (4 x^{2} + 28 x + 49)]$

Step 2.4

Since $6$ is constant with respect to $x$ , the derivative of $6 (4 x^{2} + 28 x + 49)$ with respect to $x$ is $6 \frac{d}{d x} [4 x^{2} + 28 x + 49]$ .

$6 \frac{d}{d x} [4 x^{2} + 28 x + 49]$

Step 2.5

By the Sum Rule, the derivative of $4 x^{2} + 28 x + 49$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [28 x] + \frac{d}{d x} [49]$ .

$6 (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [28 x] + \frac{d}{d x} [49])$

Step 2.6

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$6 (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [28 x] + \frac{d}{d x} [49])$

Step 2.7

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

$6 (4 (2 x) + \frac{d}{d x} [28 x] + \frac{d}{d x} [49])$

Step 2.8

Multiply $2$ by $4$ .

$6 (8 x + \frac{d}{d x} [28 x] + \frac{d}{d x} [49])$

Step 2.9

Since $28$ is constant with respect to $x$ , the derivative of $28 x$ with respect to $x$ is $28 \frac{d}{d x} [x]$ .

$6 (8 x + 28 \frac{d}{d x} [x] + \frac{d}{d x} [49])$

Step 2.10

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

$6 (8 x + 28 \cdot 1 + \frac{d}{d x} [49])$

Step 2.11

Multiply $28$ by $1$ .

$6 (8 x + 28 + \frac{d}{d x} [49])$

Step 2.12

Since $49$ is constant with respect to $x$ , the derivative of $49$ with respect to $x$ is $0$ .

$6 (8 x + 28 + 0)$

Step 2.13

Add $8 x + 28$ and $0$ .

$6 (8 x + 28)$

Step 2.14

Simplify.

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Step 2.14.1

Apply the distributive property.

$6 (8 x) + 6 \cdot 28$

Step 2.14.2

Combine terms.

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Step 2.14.2.1

Multiply $8$ by $6$ .

$48 x + 6 \cdot 28$

Step 2.14.2.2

Multiply $6$ by $28$ .

$f'' (x) = 48 x + 168$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $48 x + 168$ with respect to $x$ is $\frac{d}{d x} [48 x] + \frac{d}{d x} [168]$ .

$\frac{d}{d x} [48 x] + \frac{d}{d x} [168]$

Step 3.2

Evaluate $\frac{d}{d x} [48 x]$ .

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Step 3.2.1

Since $48$ is constant with respect to $x$ , the derivative of $48 x$ with respect to $x$ is $48 \frac{d}{d x} [x]$ .

$48 \frac{d}{d x} [x] + \frac{d}{d x} [168]$

Step 3.2.2

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

$48 \cdot 1 + \frac{d}{d x} [168]$

Step 3.2.3

Multiply $48$ by $1$ .

$48 + \frac{d}{d x} [168]$

Step 3.3

Differentiate using the Constant Rule.

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Step 3.3.1

Since $168$ is constant with respect to $x$ , the derivative of $168$ with respect to $x$ is $0$ .

$48 + 0$

Step 3.3.2

Add $48$ and $0$ .

$f''' (x) = 48$

Step 4

Since $48$ is constant with respect to $x$ , the derivative of $48$ with respect to $x$ is $0$ .

$f^{4} (x) = 0$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $0$ .

$0$