Calculus Examples

${(1 + \frac{x}{20})}^{5}$

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 $f (x) = x^{5}$ and $g (x) = 1 + \frac{x}{20}$ .

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Step 1.1.1

To apply the Chain Rule, set $u$ as $1 + \frac{x}{20}$ .

$f' (x) = \frac{d}{d u} (u^{5}) \frac{d}{d x} (1 + \frac{x}{20})$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 5$ .

$f' (x) = 5 u^{4} \frac{d}{d x} (1 + \frac{x}{20})$

Step 1.1.3

Replace all occurrences of $u$ with $1 + \frac{x}{20}$ .

$f' (x) = 5 {(1 + \frac{x}{20})}^{4} \frac{d}{d x} (1 + \frac{x}{20})$

Step 1.2

Differentiate.

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Step 1.2.1

By the Sum Rule, the derivative of $1 + \frac{x}{20}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [\frac{x}{20}]$ .

$f' (x) = 5 {(1 + \frac{x}{20})}^{4} (\frac{d}{d x} (1) + \frac{d}{d x} (\frac{x}{20}))$

Step 1.2.2

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

$f' (x) = 5 {(1 + \frac{x}{20})}^{4} (0 + \frac{d}{d x} (\frac{x}{20}))$

Step 1.2.3

Add $0$ and $\frac{d}{d x} [\frac{x}{20}]$ .

$f' (x) = 5 {(1 + \frac{x}{20})}^{4} \frac{d}{d x} (\frac{x}{20})$

Step 1.2.4

Since $\frac{1}{20}$ is constant with respect to $x$ , the derivative of $\frac{x}{20}$ with respect to $x$ is $\frac{1}{20} \frac{d}{d x} [x]$ .

$f' (x) = 5 {(1 + \frac{x}{20})}^{4} (\frac{1}{20} \cdot \frac{d}{d x} (x))$

Step 1.2.5

Simplify terms.

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Step 1.2.5.1

Combine $\frac{1}{20}$ and $5$ .

$f' (x) = \frac{5}{20} \cdot {(1 + \frac{x}{20})}^{4} \frac{d}{d x} (x)$

Step 1.2.5.2

Combine $\frac{5}{20}$ and ${(1 + \frac{x}{20})}^{4}$ .

$f' (x) = \frac{5 {(1 + \frac{x}{20})}^{4}}{20} \cdot \frac{d}{d x} (x)$

Step 1.2.5.3

Cancel the common factor of $5$ and $20$ .

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Step 1.2.5.3.1

Factor $5$ out of $5 {(1 + \frac{x}{20})}^{4}$ .

$f' (x) = \frac{5 ({(1 + \frac{x}{20})}^{4})}{20} \cdot \frac{d}{d x} (x)$

Step 1.2.5.3.2

Cancel the common factors.

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Step 1.2.5.3.2.1

Factor $5$ out of $20$ .

$f' (x) = \frac{5 {(1 + \frac{x}{20})}^{4}}{5 \cdot 4} \cdot \frac{d}{d x} (x)$

Step 1.2.5.3.2.2

Cancel the common factor.

$f' (x) = \frac{5 {(1 + \frac{x}{20})}^{4}}{5 \cdot 4} \cdot \frac{d}{d x} (x)$

Step 1.2.5.3.2.3

Rewrite the expression.

$f' (x) = \frac{{(1 + \frac{x}{20})}^{4}}{4} \cdot \frac{d}{d x} (x)$

Step 1.2.6

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

$f' (x) = \frac{{(1 + \frac{x}{20})}^{4}}{4} \cdot 1$

Step 1.2.7

Multiply $\frac{{(1 + \frac{x}{20})}^{4}}{4}$ by $1$ .

$f' (x) = \frac{{(1 + \frac{x}{20})}^{4}}{4}$

Step 1.3

Simplify.

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Step 1.3.1

Simplify the numerator.

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Step 1.3.1.1

Use the Binomial Theorem.

$f' (x) = \frac{1^{4} + 4 \cdot (1^{3} (\frac{x}{20})) + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2

Simplify each term.

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Step 1.3.1.2.1

One to any power is one.

$f' (x) = \frac{1 + 4 \cdot (1^{3} (\frac{x}{20})) + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.2

Cancel the common factor of $4$ .

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Step 1.3.1.2.2.1

Factor $4$ out of $4 \cdot 1^{3}$ .

$f' (x) = \frac{1 + 4 (1^{3}) (\frac{x}{20}) + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.2.2

Factor $4$ out of $20$ .

$f' (x) = \frac{1 + 4 (1^{3}) (\frac{x}{4 (5)}) + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.2.3

Cancel the common factor.

$f' (x) = \frac{1 + 4 \cdot (1^{3} (\frac{x}{4 \cdot 5})) + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.2.4

Rewrite the expression.

$f' (x) = \frac{1 + 1^{3} (\frac{x}{5}) + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.3

Combine $1^{3}$ and $\frac{x}{5}$ .

$f' (x) = \frac{1 + \frac{1^{3} x}{5} + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.4

Simplify the numerator.

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Step 1.3.1.2.4.1

One to any power is one.

$f' (x) = \frac{1 + \frac{1 x}{5} + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.4.2

Multiply $x$ by $1$ .

$f' (x) = \frac{1 + \frac{x}{5} + 6 \cdot (1^{2} {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.5

One to any power is one.

$f' (x) = \frac{1 + \frac{x}{5} + 6 \cdot (1 {(\frac{x}{20})}^{2}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.6

Multiply $6$ by $1$ .

$f' (x) = \frac{1 + \frac{x}{5} + 6 {(\frac{x}{20})}^{2} + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.7

Apply the product rule to $\frac{x}{20}$ .

$f' (x) = \frac{1 + \frac{x}{5} + 6 (\frac{x^{2}}{20^{2}}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.8

Raise $20$ to the power of $2$ .

$f' (x) = \frac{1 + \frac{x}{5} + 6 (\frac{x^{2}}{400}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.9

Cancel the common factor of $2$ .

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Step 1.3.1.2.9.1

Factor $2$ out of $6$ .

$f' (x) = \frac{1 + \frac{x}{5} + 2 (3) (\frac{x^{2}}{400}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.9.2

Factor $2$ out of $400$ .

$f' (x) = \frac{1 + \frac{x}{5} + 2 \cdot (3 (\frac{x^{2}}{2 \cdot 200})) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.9.3

Cancel the common factor.

$f' (x) = \frac{1 + \frac{x}{5} + 2 \cdot (3 (\frac{x^{2}}{2 \cdot 200})) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.9.4

Rewrite the expression.

$f' (x) = \frac{1 + \frac{x}{5} + 3 (\frac{x^{2}}{200}) + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.10

Combine $3$ and $\frac{x^{2}}{200}$ .

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + 4 \cdot (1 {(\frac{x}{20})}^{3}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.11

Multiply $4$ by $1$ .

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + 4 {(\frac{x}{20})}^{3} + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.12

Apply the product rule to $\frac{x}{20}$ .

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + 4 (\frac{x^{3}}{20^{3}}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.13

Raise $20$ to the power of $3$ .

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + 4 (\frac{x^{3}}{8000}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.14

Cancel the common factor of $4$ .

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Step 1.3.1.2.14.1

Factor $4$ out of $8000$ .

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + 4 (\frac{x^{3}}{4 (2000)}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.14.2

Cancel the common factor.

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + 4 (\frac{x^{3}}{4 \cdot 2000}) + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.14.3

Rewrite the expression.

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + \frac{x^{3}}{2000} + {(\frac{x}{20})}^{4}}{4}$

Step 1.3.1.2.15

Apply the product rule to $\frac{x}{20}$ .

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + \frac{x^{3}}{2000} + \frac{x^{4}}{20^{4}}}{4}$

Step 1.3.1.2.16

Raise $20$ to the power of $4$ .

$f' (x) = \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + \frac{x^{3}}{2000} + \frac{x^{4}}{160000}}{4}$

Step 1.3.2

Combine terms.

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Step 1.3.2.1

Multiply $\frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + \frac{x^{3}}{2000} + \frac{x^{4}}{160000}}{4}$ by $\frac{160000}{160000}$ .

$f' (x) = \frac{160000}{160000} \cdot \frac{1 + \frac{x}{5} + \frac{3 x^{2}}{200} + \frac{x^{3}}{2000} + \frac{x^{4}}{160000}}{4}$

Step 1.3.2.2

Combine.

$f' (x) = \frac{160000 (1 + \frac{x}{5} + \frac{3 x^{2}}{200} + \frac{x^{3}}{2000} + \frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.3

Apply the distributive property.

$f' (x) = \frac{160000 \cdot 1 + 160000 (\frac{x}{5}) + 160000 (\frac{3 x^{2}}{200}) + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.4

Cancel the common factor of $5$ .

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Step 1.3.2.4.1

Factor $5$ out of $160000$ .

$f' (x) = \frac{160000 \cdot 1 + 5 (32000) (\frac{x}{5}) + 160000 (\frac{3 x^{2}}{200}) + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.4.2

Cancel the common factor.

$f' (x) = \frac{160000 \cdot 1 + 5 \cdot (32000 (\frac{x}{5})) + 160000 (\frac{3 x^{2}}{200}) + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.4.3

Rewrite the expression.

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 160000 (\frac{3 x^{2}}{200}) + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.5

Cancel the common factor of $200$ .

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Step 1.3.2.5.1

Factor $200$ out of $160000$ .

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 200 (800) (\frac{3 x^{2}}{200}) + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.5.2

Cancel the common factor.

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 200 \cdot (800 (\frac{3 x^{2}}{200})) + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.5.3

Rewrite the expression.

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 800 (3 x^{2}) + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.6

Multiply $3$ by $800$ .

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 2400 x^{2} + 160000 (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.7

Cancel the common factor of $2000$ .

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Step 1.3.2.7.1

Factor $2000$ out of $160000$ .

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 2400 x^{2} + 2000 (80) (\frac{x^{3}}{2000}) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.7.2

Cancel the common factor.

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 2400 x^{2} + 2000 \cdot (80 (\frac{x^{3}}{2000})) + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.7.3

Rewrite the expression.

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 2400 x^{2} + 80 x^{3} + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.8

Cancel the common factor of $160000$ .

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Step 1.3.2.8.1

Cancel the common factor.

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 2400 x^{2} + 80 x^{3} + 160000 (\frac{x^{4}}{160000})}{160000 \cdot 4}$

Step 1.3.2.8.2

Rewrite the expression.

$f' (x) = \frac{160000 \cdot 1 + 32000 x + 2400 x^{2} + 80 x^{3} + x^{4}}{160000 \cdot 4}$

Step 1.3.2.9

Multiply $160000$ by $1$ .

$f' (x) = \frac{160000 + 32000 x + 2400 x^{2} + 80 x^{3} + x^{4}}{160000 \cdot 4}$

Step 1.3.2.10

Multiply $160000$ by $4$ .

$f' (x) = \frac{160000 + 32000 x + 2400 x^{2} + 80 x^{3} + x^{4}}{640000}$

Step 1.3.3

Reorder terms.

$f' (x) = \frac{x^{4} + 80 x^{3} + 2400 x^{2} + 32000 x + 160000}{640000}$

Step 2

Find the second derivative.

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Step 2.1

Since $\frac{1}{640000}$ is constant with respect to $x$ , the derivative of $\frac{x^{4} + 80 x^{3} + 2400 x^{2} + 32000 x + 160000}{640000}$ with respect to $x$ is $\frac{1}{640000} \frac{d}{d x} [x^{4} + 80 x^{3} + 2400 x^{2} + 32000 x + 160000]$ .

$f'' (x) = \frac{1}{640000} \cdot \frac{d}{d x} (x^{4} + 80 x^{3} + 2400 x^{2} + 32000 x + 160000)$

Step 2.2

By the Sum Rule, the derivative of $x^{4} + 80 x^{3} + 2400 x^{2} + 32000 x + 160000$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [80 x^{3}] + \frac{d}{d x} [2400 x^{2}] + \frac{d}{d x} [32000 x] + \frac{d}{d x} [160000]$ .

$f'' (x) = \frac{1}{640000} \cdot (\frac{d}{d x} (x^{4}) + \frac{d}{d x} (80 x^{3}) + \frac{d}{d x} (2400 x^{2}) + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.3

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

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + \frac{d}{d x} (80 x^{3}) + \frac{d}{d x} (2400 x^{2}) + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.4

Since $80$ is constant with respect to $x$ , the derivative of $80 x^{3}$ with respect to $x$ is $80 \frac{d}{d x} [x^{3}]$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 80 \frac{d}{d x} (x^{3}) + \frac{d}{d x} (2400 x^{2}) + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.5

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

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 80 (3 x^{2}) + \frac{d}{d x} (2400 x^{2}) + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.6

Multiply $3$ by $80$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + \frac{d}{d x} (2400 x^{2}) + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.7

Since $2400$ is constant with respect to $x$ , the derivative of $2400 x^{2}$ with respect to $x$ is $2400 \frac{d}{d x} [x^{2}]$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 2400 \frac{d}{d x} (x^{2}) + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.8

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

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 2400 (2 x) + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.9

Multiply $2$ by $2400$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 4800 x + \frac{d}{d x} (32000 x) + \frac{d}{d x} (160000))$

Step 2.10

Since $32000$ is constant with respect to $x$ , the derivative of $32000 x$ with respect to $x$ is $32000 \frac{d}{d x} [x]$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 4800 x + 32000 \frac{d}{d x} (x) + \frac{d}{d x} (160000))$

Step 2.11

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

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 4800 x + 32000 \cdot 1 + \frac{d}{d x} (160000))$

Step 2.12

Multiply $32000$ by $1$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 4800 x + 32000 + \frac{d}{d x} (160000))$

Step 2.13

Since $160000$ is constant with respect to $x$ , the derivative of $160000$ with respect to $x$ is $0$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 4800 x + 32000 + 0)$

Step 2.14

Add $4 x^{3} + 240 x^{2} + 4800 x + 32000$ and $0$ .

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3} + 240 x^{2} + 4800 x + 32000)$

Step 2.15

Simplify.

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Step 2.15.1

Apply the distributive property.

$f'' (x) = \frac{1}{640000} \cdot (4 x^{3}) + \frac{1}{640000} \cdot (240 x^{2}) + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2

Combine terms.

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Step 2.15.2.1

Combine $4$ and $\frac{1}{640000}$ .

$f'' (x) = \frac{4}{640000} x^{3} + \frac{1}{640000} \cdot (240 x^{2}) + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.2

Combine $\frac{4}{640000}$ and $x^{3}$ .

$f'' (x) = \frac{4 x^{3}}{640000} + \frac{1}{640000} \cdot (240 x^{2}) + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.3

Cancel the common factor of $4$ and $640000$ .

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Step 2.15.2.3.1

Factor $4$ out of $4 x^{3}$ .

$f'' (x) = \frac{4 (x^{3})}{640000} + \frac{1}{640000} \cdot (240 x^{2}) + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.3.2

Cancel the common factors.

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Step 2.15.2.3.2.1

Factor $4$ out of $640000$ .

$f'' (x) = \frac{4 x^{3}}{4 \cdot 160000} + \frac{1}{640000} \cdot (240 x^{2}) + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.3.2.2

Cancel the common factor.

$f'' (x) = \frac{4 x^{3}}{4 \cdot 160000} + \frac{1}{640000} \cdot (240 x^{2}) + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.3.2.3

Rewrite the expression.

$f'' (x) = \frac{x^{3}}{160000} + \frac{1}{640000} \cdot (240 x^{2}) + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.4

Combine $240$ and $\frac{1}{640000}$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{240}{640000} x^{2} + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.5

Combine $\frac{240}{640000}$ and $x^{2}$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{240 x^{2}}{640000} + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.6

Cancel the common factor of $240$ and $640000$ .

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Step 2.15.2.6.1

Factor $80$ out of $240 x^{2}$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{80 (3 x^{2})}{640000} + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.6.2

Cancel the common factors.

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Step 2.15.2.6.2.1

Factor $80$ out of $640000$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{80 (3 x^{2})}{80 (8000)} + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.6.2.2

Cancel the common factor.

$f'' (x) = \frac{x^{3}}{160000} + \frac{80 (3 x^{2})}{80 \cdot 8000} + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.6.2.3

Rewrite the expression.

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{1}{640000} \cdot (4800 x) + \frac{1}{640000} \cdot 32000$

Step 2.15.2.7

Combine $4800$ and $\frac{1}{640000}$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{4800}{640000} x + \frac{1}{640000} \cdot 32000$

Step 2.15.2.8

Combine $\frac{4800}{640000}$ and $x$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{4800 x}{640000} + \frac{1}{640000} \cdot 32000$

Step 2.15.2.9

Cancel the common factor of $4800$ and $640000$ .

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Step 2.15.2.9.1

Factor $1600$ out of $4800 x$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{1600 (3 x)}{640000} + \frac{1}{640000} \cdot 32000$

Step 2.15.2.9.2

Cancel the common factors.

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Step 2.15.2.9.2.1

Factor $1600$ out of $640000$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{1600 (3 x)}{1600 (400)} + \frac{1}{640000} \cdot 32000$

Step 2.15.2.9.2.2

Cancel the common factor.

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{1600 (3 x)}{1600 \cdot 400} + \frac{1}{640000} \cdot 32000$

Step 2.15.2.9.2.3

Rewrite the expression.

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{3 x}{400} + \frac{1}{640000} \cdot 32000$

Step 2.15.2.10

Combine $\frac{1}{640000}$ and $32000$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{3 x}{400} + \frac{32000}{640000}$

Step 2.15.2.11

Cancel the common factor of $32000$ and $640000$ .

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Step 2.15.2.11.1

Factor $32000$ out of $32000$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{3 x}{400} + \frac{32000 (1)}{640000}$

Step 2.15.2.11.2

Cancel the common factors.

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Step 2.15.2.11.2.1

Factor $32000$ out of $640000$ .

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{3 x}{400} + \frac{32000 \cdot 1}{32000 \cdot 20}$

Step 2.15.2.11.2.2

Cancel the common factor.

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{3 x}{400} + \frac{32000 \cdot 1}{32000 \cdot 20}$

Step 2.15.2.11.2.3

Rewrite the expression.

$f'' (x) = \frac{x^{3}}{160000} + \frac{3 x^{2}}{8000} + \frac{3 x}{400} + \frac{1}{20}$