Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{3}{20} x^{- 4} + x^{5} + \frac{1}{3} x^{3} + x^{2}$ with respect to $x$ is $\frac{d}{d x} [\frac{3}{20} x^{- 4}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$ .

$\frac{d}{d x} [\frac{3}{20} x^{- 4}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{3}{20} x^{- 4}]$ .

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Step 1.2.1

Since $\frac{3}{20}$ is constant with respect to $x$ , the derivative of $\frac{3}{20} x^{- 4}$ with respect to $x$ is $\frac{3}{20} \frac{d}{d x} [x^{- 4}]$ .

$\frac{3}{20} \frac{d}{d x} [x^{- 4}] + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.2

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

$\frac{3}{20} (- 4 x^{- 5}) + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.3

Combine $- 4$ and $\frac{3}{20}$ .

$\frac{- 4 \cdot 3}{20} x^{- 5} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.4

Multiply $- 4$ by $3$ .

$\frac{- 12}{20} x^{- 5} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.5

Combine $\frac{- 12}{20}$ and $x^{- 5}$ .

$\frac{- 12 x^{- 5}}{20} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.6

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{- 12}{20 x^{5}} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.7

Cancel the common factor of $- 12$ and $20$ .

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Step 1.2.7.1

Factor $4$ out of $- 12$ .

$\frac{4 (- 3)}{20 x^{5}} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

Factor $4$ out of $20 x^{5}$ .

$\frac{4 (- 3)}{4 (5 x^{5})} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.7.2.2

Cancel the common factor.

$\frac{4 \cdot - 3}{4 (5 x^{5})} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.7.2.3

Rewrite the expression.

$\frac{- 3}{5 x^{5}} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.2.8

Move the negative in front of the fraction.

$- \frac{3}{5 x^{5}} + \frac{d}{d x} [x^{5}] + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.3

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

$- \frac{3}{5 x^{5}} + 5 x^{4} + \frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.4

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 1.4.1

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

$- \frac{3}{5 x^{5}} + 5 x^{4} + \frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [x^{2}]$

Step 1.4.2

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

$- \frac{3}{5 x^{5}} + 5 x^{4} + \frac{1}{3} (3 x^{2}) + \frac{d}{d x} [x^{2}]$

Step 1.4.3

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

$- \frac{3}{5 x^{5}} + 5 x^{4} + \frac{3}{3} x^{2} + \frac{d}{d x} [x^{2}]$

Step 1.4.4

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

$- \frac{3}{5 x^{5}} + 5 x^{4} + \frac{3 x^{2}}{3} + \frac{d}{d x} [x^{2}]$

Step 1.4.5

Cancel the common factor of $3$ .

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Step 1.4.5.1

Cancel the common factor.

$- \frac{3}{5 x^{5}} + 5 x^{4} + \frac{3 x^{2}}{3} + \frac{d}{d x} [x^{2}]$

Step 1.4.5.2

Divide $x^{2}$ by $1$ .

$- \frac{3}{5 x^{5}} + 5 x^{4} + x^{2} + \frac{d}{d x} [x^{2}]$

Step 1.5

Differentiate using the Power Rule.

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Step 1.5.1

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

$- \frac{3}{5 x^{5}} + 5 x^{4} + x^{2} + 2 x$

Step 1.5.2

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $5 x^{4} + x^{2} + 2 x - \frac{3}{5 x^{5}}$ with respect to $x$ is $\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$ .

$\frac{d}{d x} [5 x^{4}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.2

Evaluate $\frac{d}{d x} [5 x^{4}]$ .

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Step 2.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4}]$ .

$5 \frac{d}{d x} [x^{4}] + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.2.2

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

$5 (4 x^{3}) + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.2.3

Multiply $4$ by $5$ .

$20 x^{3} + \frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.3

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

$20 x^{3} + 2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.4

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

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Step 2.4.1

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

$20 x^{3} + 2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.4.2

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

$20 x^{3} + 2 x + 2 \cdot 1 + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.4.3

Multiply $2$ by $1$ .

$20 x^{3} + 2 x + 2 + \frac{d}{d x} [- \frac{3}{5 x^{5}}]$

Step 2.5

Evaluate $\frac{d}{d x} [- \frac{3}{5 x^{5}}]$ .

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Step 2.5.1

Since $- \frac{3}{5}$ is constant with respect to $x$ , the derivative of $- \frac{3}{5 x^{5}}$ with respect to $x$ is $- \frac{3}{5} \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} \frac{d}{d x} [\frac{1}{x^{5}}]$

Step 2.5.2

Rewrite $\frac{1}{x^{5}}$ as ${(x^{5})}^{- 1}$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} \frac{d}{d x} [{(x^{5})}^{- 1}]$

Step 2.5.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = x^{5}$ .

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Step 2.5.3.1

To apply the Chain Rule, set $u$ as $x^{5}$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{5}])$

Step 2.5.3.2

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

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- u^{- 2} \frac{d}{d x} [x^{5}])$

Step 2.5.3.3

Replace all occurrences of $u$ with $x^{5}$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}])$

Step 2.5.4

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

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- {(x^{5})}^{- 2} (5 x^{4}))$

Step 2.5.5

Multiply the exponents in ${(x^{5})}^{- 2}$ .

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Step 2.5.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- x^{5 \cdot - 2} (5 x^{4}))$

Step 2.5.5.2

Multiply $5$ by $- 2$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- x^{- 10} (5 x^{4}))$

Step 2.5.6

Multiply $5$ by $- 1$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- 5 x^{- 10} x^{4})$

Step 2.5.7

Multiply $x^{- 10}$ by $x^{4}$ by adding the exponents.

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Step 2.5.7.1

Move $x^{4}$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- 5 (x^{4} x^{- 10}))$

Step 2.5.7.2

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

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- 5 x^{4 - 10})$

Step 2.5.7.3

Subtract $10$ from $4$ .

$20 x^{3} + 2 x + 2 - \frac{3}{5} (- 5 x^{- 6})$

Step 2.5.8

Multiply $- 5$ by $- 1$ .

$20 x^{3} + 2 x + 2 + 5 (\frac{3}{5}) x^{- 6}$

Step 2.5.9

Combine $5$ and $\frac{3}{5}$ .

$20 x^{3} + 2 x + 2 + \frac{5 \cdot 3}{5} x^{- 6}$

Step 2.5.10

Multiply $5$ by $3$ .

$20 x^{3} + 2 x + 2 + \frac{15}{5} x^{- 6}$

Step 2.5.11

Combine $\frac{15}{5}$ and $x^{- 6}$ .

$20 x^{3} + 2 x + 2 + \frac{15 x^{- 6}}{5}$

Step 2.5.12

Move $x^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$20 x^{3} + 2 x + 2 + \frac{15}{5 x^{6}}$

Step 2.5.13

Cancel the common factor of $15$ and $5$ .

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Step 2.5.13.1

Factor $5$ out of $15$ .

$20 x^{3} + 2 x + 2 + \frac{5 \cdot 3}{5 x^{6}}$

Step 2.5.13.2

Cancel the common factors.

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Step 2.5.13.2.1

Factor $5$ out of $5 x^{6}$ .

$20 x^{3} + 2 x + 2 + \frac{5 \cdot 3}{5 (x^{6})}$

Step 2.5.13.2.2

Cancel the common factor.

$20 x^{3} + 2 x + 2 + \frac{5 \cdot 3}{5 x^{6}}$

Step 2.5.13.2.3

Rewrite the expression.

$20 x^{3} + 2 x + 2 + \frac{3}{x^{6}}$

Step 2.6

Reorder terms.

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

Step 3

The second derivative of $f (x)$ with respect to $x$ is $20 x^{3} + 2 x + \frac{3}{x^{6}} + 2$ .

$20 x^{3} + 2 x + \frac{3}{x^{6}} + 2$