Calculus Examples

$f (x) = 3 x {(x + 4)}^{2}$

Step 1

Find the first derivative.

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Step 1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$\frac{d}{d x} [3 x ((x + 4) (x + 4))]$

Step 1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\frac{d}{d x} [3 x (x (x + 4) + 4 (x + 4))]$

Step 1.2.2

Apply the distributive property.

$\frac{d}{d x} [3 x (x \cdot x + x \cdot 4 + 4 (x + 4))]$

Step 1.2.3

Apply the distributive property.

$\frac{d}{d x} [3 x (x \cdot x + x \cdot 4 + 4 x + 4 \cdot 4)]$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$\frac{d}{d x} [3 x (x^{2} + x \cdot 4 + 4 x + 4 \cdot 4)]$

Step 1.3.1.2

Move $4$ to the left of $x$ .

$\frac{d}{d x} [3 x (x^{2} + 4 \cdot x + 4 x + 4 \cdot 4)]$

Step 1.3.1.3

Multiply $4$ by $4$ .

$\frac{d}{d x} [3 x (x^{2} + 4 x + 4 x + 16)]$

Step 1.3.2

Add $4 x$ and $4 x$ .

$\frac{d}{d x} [3 x (x^{2} + 8 x + 16)]$

Step 1.4

Since $3$ is constant with respect to $x$ , the derivative of $3 x (x^{2} + 8 x + 16)$ with respect to $x$ is $3 \frac{d}{d x} [x (x^{2} + 8 x + 16)]$ .

$3 \frac{d}{d x} [x (x^{2} + 8 x + 16)]$

Step 1.5

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 $g (x) = x^{2} + 8 x + 16$ .

$3 (x \frac{d}{d x} [x^{2} + 8 x + 16] + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6

Differentiate.

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Step 1.6.1

By the Sum Rule, the derivative of $x^{2} + 8 x + 16$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [16]$ .

$3 (x (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x] + \frac{d}{d x} [16]) + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6.2

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

$3 (x (2 x + \frac{d}{d x} [8 x] + \frac{d}{d x} [16]) + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6.3

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$3 (x (2 x + 8 \frac{d}{d x} [x] + \frac{d}{d x} [16]) + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6.4

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

$3 (x (2 x + 8 \cdot 1 + \frac{d}{d x} [16]) + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6.5

Multiply $8$ by $1$ .

$3 (x (2 x + 8 + \frac{d}{d x} [16]) + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6.6

Since $16$ is constant with respect to $x$ , the derivative of $16$ with respect to $x$ is $0$ .

$3 (x (2 x + 8 + 0) + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6.7

Add $2 x + 8$ and $0$ .

$3 (x (2 x + 8) + (x^{2} + 8 x + 16) \frac{d}{d x} [x])$

Step 1.6.8

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

$3 (x (2 x + 8) + (x^{2} + 8 x + 16) \cdot 1)$

Step 1.6.9

Multiply $x^{2} + 8 x + 16$ by $1$ .

$3 (x (2 x + 8) + x^{2} + 8 x + 16)$

Step 1.7

Simplify.

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Step 1.7.1

Apply the distributive property.

$3 (x (2 x) + x \cdot 8 + x^{2} + 8 x + 16)$

Step 1.7.2

Apply the distributive property.

$3 (x (2 x)) + 3 (x \cdot 8) + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3

Combine terms.

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Step 1.7.3.1

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

$3 (2 (x^{1} x)) + 3 (x \cdot 8) + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.2

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

$3 (2 (x^{1} x^{1})) + 3 (x \cdot 8) + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.3

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

$3 (2 x^{1 + 1}) + 3 (x \cdot 8) + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.4

Add $1$ and $1$ .

$3 (2 x^{2}) + 3 (x \cdot 8) + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.5

Multiply $2$ by $3$ .

$6 x^{2} + 3 (x \cdot 8) + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.6

Move $8$ to the left of $x$ .

$6 x^{2} + 3 (8 \cdot x) + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.7

Multiply $8$ by $3$ .

$6 x^{2} + 24 x + 3 x^{2} + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.8

Add $6 x^{2}$ and $3 x^{2}$ .

$9 x^{2} + 24 x + 3 (8 x) + 3 \cdot 16$

Step 1.7.3.9

Multiply $8$ by $3$ .

$9 x^{2} + 24 x + 24 x + 3 \cdot 16$

Step 1.7.3.10

Add $24 x$ and $24 x$ .

$9 x^{2} + 48 x + 3 \cdot 16$

Step 1.7.3.11

Multiply $3$ by $16$ .

$f' (x) = 9 x^{2} + 48 x + 48$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $9 x^{2} + 48 x + 48$ with respect to $x$ is $\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [48 x] + \frac{d}{d x} [48]$ .

$\frac{d}{d x} [9 x^{2}] + \frac{d}{d x} [48 x] + \frac{d}{d x} [48]$

Step 2.2

Evaluate $\frac{d}{d x} [9 x^{2}]$ .

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Step 2.2.1

Since $9$ is constant with respect to $x$ , the derivative of $9 x^{2}$ with respect to $x$ is $9 \frac{d}{d x} [x^{2}]$ .

$9 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [48 x] + \frac{d}{d x} [48]$

Step 2.2.2

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

$9 (2 x) + \frac{d}{d x} [48 x] + \frac{d}{d x} [48]$

Step 2.2.3

Multiply $2$ by $9$ .

$18 x + \frac{d}{d x} [48 x] + \frac{d}{d x} [48]$

Step 2.3

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

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Step 2.3.1

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

$18 x + 48 \frac{d}{d x} [x] + \frac{d}{d x} [48]$

Step 2.3.2

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

$18 x + 48 \cdot 1 + \frac{d}{d x} [48]$

Step 2.3.3

Multiply $48$ by $1$ .

$18 x + 48 + \frac{d}{d x} [48]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

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

$18 x + 48 + 0$

Step 2.4.2

Add $18 x + 48$ and $0$ .

$f'' (x) = 18 x + 48$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $18 x + 48$ .

$18 x + 48$