Calculus Examples

$f (x) = {(8 x + 9)}^{4}$

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 $f (x) = x^{4}$ and $g (x) = 8 x + 9$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $8 x + 9$ .

$\frac{d}{d u} [u^{4}] \frac{d}{d x} [8 x + 9]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 4$ .

$4 u^{3} \frac{d}{d x} [8 x + 9]$

Step 1.1.3

Replace all occurrences of $u$ with $8 x + 9$ .

$4 {(8 x + 9)}^{3} \frac{d}{d x} [8 x + 9]$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $8 x + 9$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [9]$ .

$4 {(8 x + 9)}^{3} (\frac{d}{d x} [8 x] + \frac{d}{d x} [9])$

Step 1.2.2

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

$4 {(8 x + 9)}^{3} (8 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 1.2.3

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

$4 {(8 x + 9)}^{3} (8 \cdot 1 + \frac{d}{d x} [9])$

Step 1.2.4

Multiply $8$ by $1$ .

$4 {(8 x + 9)}^{3} (8 + \frac{d}{d x} [9])$

Step 1.2.5

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$4 {(8 x + 9)}^{3} (8 + 0)$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Add $8$ and $0$ .

$4 {(8 x + 9)}^{3} \cdot 8$

Step 1.2.6.2

Multiply $8$ by $4$ .

$f' (x) = 32 {(8 x + 9)}^{3}$

Step 2

Find the second derivative.

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Step 2.1

Since $32$ is constant with respect to $x$ , the derivative of $32 {(8 x + 9)}^{3}$ with respect to $x$ is $32 \frac{d}{d x} [{(8 x + 9)}^{3}]$ .

$32 \frac{d}{d x} [{(8 x + 9)}^{3}]$

Step 2.2

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 $g (x) = 8 x + 9$ .

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Step 2.2.1

To apply the Chain Rule, set $u$ as $8 x + 9$ .

$32 (\frac{d}{d u} [u^{3}] \frac{d}{d x} [8 x + 9])$

Step 2.2.2

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

$32 (3 u^{2} \frac{d}{d x} [8 x + 9])$

Step 2.2.3

Replace all occurrences of $u$ with $8 x + 9$ .

$32 (3 {(8 x + 9)}^{2} \frac{d}{d x} [8 x + 9])$

Step 2.3

Differentiate.

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Step 2.3.1

Multiply $3$ by $32$ .

$96 ({(8 x + 9)}^{2} \frac{d}{d x} [8 x + 9])$

Step 2.3.2

By the Sum Rule, the derivative of $8 x + 9$ with respect to $x$ is $\frac{d}{d x} [8 x] + \frac{d}{d x} [9]$ .

$96 {(8 x + 9)}^{2} (\frac{d}{d x} [8 x] + \frac{d}{d x} [9])$

Step 2.3.3

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

$96 {(8 x + 9)}^{2} (8 \frac{d}{d x} [x] + \frac{d}{d x} [9])$

Step 2.3.4

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

$96 {(8 x + 9)}^{2} (8 \cdot 1 + \frac{d}{d x} [9])$

Step 2.3.5

Multiply $8$ by $1$ .

$96 {(8 x + 9)}^{2} (8 + \frac{d}{d x} [9])$

Step 2.3.6

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$96 {(8 x + 9)}^{2} (8 + 0)$

Step 2.3.7

Simplify the expression.

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Step 2.3.7.1

Add $8$ and $0$ .

$96 {(8 x + 9)}^{2} \cdot 8$

Step 2.3.7.2

Multiply $8$ by $96$ .

$f'' (x) = 768 {(8 x + 9)}^{2}$

Step 3

Find the third derivative.

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Step 3.1

Rewrite ${(8 x + 9)}^{2}$ as $(8 x + 9) (8 x + 9)$ .

$\frac{d}{d x} [768 ((8 x + 9) (8 x + 9))]$

Step 3.2

Expand $(8 x + 9) (8 x + 9)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

$\frac{d}{d x} [768 (8 x (8 x + 9) + 9 (8 x + 9))]$

Step 3.2.2

Apply the distributive property.

$\frac{d}{d x} [768 (8 x (8 x) + 8 x \cdot 9 + 9 (8 x + 9))]$

Step 3.2.3

Apply the distributive property.

$\frac{d}{d x} [768 (8 x (8 x) + 8 x \cdot 9 + 9 (8 x) + 9 \cdot 9)]$

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Rewrite using the commutative property of multiplication.

$\frac{d}{d x} [768 (8 \cdot 8 x \cdot x + 8 x \cdot 9 + 9 (8 x) + 9 \cdot 9)]$

Step 3.3.1.2

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

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Step 3.3.1.2.1

Move $x$ .

$\frac{d}{d x} [768 (8 \cdot 8 (x \cdot x) + 8 x \cdot 9 + 9 (8 x) + 9 \cdot 9)]$

Step 3.3.1.2.2

Multiply $x$ by $x$ .

$\frac{d}{d x} [768 (8 \cdot 8 x^{2} + 8 x \cdot 9 + 9 (8 x) + 9 \cdot 9)]$

Step 3.3.1.3

Multiply $8$ by $8$ .

$\frac{d}{d x} [768 (64 x^{2} + 8 x \cdot 9 + 9 (8 x) + 9 \cdot 9)]$

Step 3.3.1.4

Multiply $9$ by $8$ .

$\frac{d}{d x} [768 (64 x^{2} + 72 x + 9 (8 x) + 9 \cdot 9)]$

Step 3.3.1.5

Multiply $8$ by $9$ .

$\frac{d}{d x} [768 (64 x^{2} + 72 x + 72 x + 9 \cdot 9)]$

Step 3.3.1.6

Multiply $9$ by $9$ .

$\frac{d}{d x} [768 (64 x^{2} + 72 x + 72 x + 81)]$

Step 3.3.2

Add $72 x$ and $72 x$ .

$\frac{d}{d x} [768 (64 x^{2} + 144 x + 81)]$

Step 3.4

Since $768$ is constant with respect to $x$ , the derivative of $768 (64 x^{2} + 144 x + 81)$ with respect to $x$ is $768 \frac{d}{d x} [64 x^{2} + 144 x + 81]$ .

$768 \frac{d}{d x} [64 x^{2} + 144 x + 81]$

Step 3.5

By the Sum Rule, the derivative of $64 x^{2} + 144 x + 81$ with respect to $x$ is $\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [144 x] + \frac{d}{d x} [81]$ .

$768 (\frac{d}{d x} [64 x^{2}] + \frac{d}{d x} [144 x] + \frac{d}{d x} [81])$

Step 3.6

Since $64$ is constant with respect to $x$ , the derivative of $64 x^{2}$ with respect to $x$ is $64 \frac{d}{d x} [x^{2}]$ .

$768 (64 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [144 x] + \frac{d}{d x} [81])$

Step 3.7

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

$768 (64 (2 x) + \frac{d}{d x} [144 x] + \frac{d}{d x} [81])$

Step 3.8

Multiply $2$ by $64$ .

$768 (128 x + \frac{d}{d x} [144 x] + \frac{d}{d x} [81])$

Step 3.9

Since $144$ is constant with respect to $x$ , the derivative of $144 x$ with respect to $x$ is $144 \frac{d}{d x} [x]$ .

$768 (128 x + 144 \frac{d}{d x} [x] + \frac{d}{d x} [81])$

Step 3.10

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

$768 (128 x + 144 \cdot 1 + \frac{d}{d x} [81])$

Step 3.11

Multiply $144$ by $1$ .

$768 (128 x + 144 + \frac{d}{d x} [81])$

Step 3.12

Since $81$ is constant with respect to $x$ , the derivative of $81$ with respect to $x$ is $0$ .

$768 (128 x + 144 + 0)$

Step 3.13

Add $128 x + 144$ and $0$ .

$768 (128 x + 144)$

Step 3.14

Simplify.

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Step 3.14.1

Apply the distributive property.

$768 (128 x) + 768 \cdot 144$

Step 3.14.2

Combine terms.

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Step 3.14.2.1

Multiply $128$ by $768$ .

$98304 x + 768 \cdot 144$

Step 3.14.2.2

Multiply $768$ by $144$ .

$f''' (x) = 98304 x + 110592$

Step 4

Find the fourth derivative.

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Step 4.1

By the Sum Rule, the derivative of $98304 x + 110592$ with respect to $x$ is $\frac{d}{d x} [98304 x] + \frac{d}{d x} [110592]$ .

$\frac{d}{d x} [98304 x] + \frac{d}{d x} [110592]$

Step 4.2

Evaluate $\frac{d}{d x} [98304 x]$ .

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Step 4.2.1

Since $98304$ is constant with respect to $x$ , the derivative of $98304 x$ with respect to $x$ is $98304 \frac{d}{d x} [x]$ .

$98304 \frac{d}{d x} [x] + \frac{d}{d x} [110592]$

Step 4.2.2

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

$98304 \cdot 1 + \frac{d}{d x} [110592]$

Step 4.2.3

Multiply $98304$ by $1$ .

$98304 + \frac{d}{d x} [110592]$

Step 4.3

Differentiate using the Constant Rule.

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Step 4.3.1

Since $110592$ is constant with respect to $x$ , the derivative of $110592$ with respect to $x$ is $0$ .

$98304 + 0$

Step 4.3.2

Add $98304$ and $0$ .

$f^{4} (x) = 98304$

Step 5

The fourth derivative of $f (x)$ with respect to $x$ is $98304$ .

$98304$