Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{5}{2} x^{2}]$ .

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Step 1.2.1

Since $- \frac{5}{2}$ is constant with respect to $x$ , the derivative of $- \frac{5}{2} x^{2}$ with respect to $x$ is $- \frac{5}{2} \frac{d}{d x} [x^{2}]$ .

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $- 1$ .

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

Step 1.2.4

Combine $- 2$ and $\frac{5}{2}$ .

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

Step 1.2.5

Multiply $- 2$ by $5$ .

$\frac{- 10}{2} x + \frac{d}{d x} [- \frac{3}{2} x^{- 3}] + \frac{d}{d x} [2 x^{3}]$

Step 1.2.6

Combine $\frac{- 10}{2}$ and $x$ .

$\frac{- 10 x}{2} + \frac{d}{d x} [- \frac{3}{2} x^{- 3}] + \frac{d}{d x} [2 x^{3}]$

Step 1.2.7

Cancel the common factor of $- 10$ and $2$ .

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Step 1.2.7.1

Factor $2$ out of $- 10 x$ .

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

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

Factor $2$ out of $2$ .

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

Step 1.2.7.2.2

Cancel the common factor.

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

Step 1.2.7.2.3

Rewrite the expression.

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

Step 1.2.7.2.4

Divide $- 5 x$ by $1$ .

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

Step 1.3

Evaluate $\frac{d}{d x} [- \frac{3}{2} x^{- 3}]$ .

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Step 1.3.1

Since $- \frac{3}{2}$ is constant with respect to $x$ , the derivative of $- \frac{3}{2} x^{- 3}$ with respect to $x$ is $- \frac{3}{2} \frac{d}{d x} [x^{- 3}]$ .

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 3$ by $- 1$ .

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

Step 1.3.4

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

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

Step 1.3.5

Multiply $3$ by $3$ .

$- 5 x + \frac{9}{2} x^{- 4} + \frac{d}{d x} [2 x^{3}]$

Step 1.3.6

Combine $\frac{9}{2}$ and $x^{- 4}$ .

$- 5 x + \frac{9 x^{- 4}}{2} + \frac{d}{d x} [2 x^{3}]$

Step 1.3.7

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 x + \frac{9}{2 x^{4}} + \frac{d}{d x} [2 x^{3}]$

Step 1.4

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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Step 1.4.1

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

$- 5 x + \frac{9}{2 x^{4}} + 2 \frac{d}{d x} [x^{3}]$

Step 1.4.2

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

$- 5 x + \frac{9}{2 x^{4}} + 2 (3 x^{2})$

Step 1.4.3

Multiply $3$ by $2$ .

$- 5 x + \frac{9}{2 x^{4}} + 6 x^{2}$

Step 1.5

Reorder terms.

$f' (x) = 6 x^{2} - 5 x + \frac{9}{2 x^{4}}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $6 x^{2} - 5 x + \frac{9}{2 x^{4}}$ with respect to $x$ is $\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [\frac{9}{2 x^{4}}]$ .

$\frac{d}{d x} [6 x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [\frac{9}{2 x^{4}}]$

Step 2.2

Evaluate $\frac{d}{d x} [6 x^{2}]$ .

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Step 2.2.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x^{2}$ with respect to $x$ is $6 \frac{d}{d x} [x^{2}]$ .

$6 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [\frac{9}{2 x^{4}}]$

Step 2.2.2

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

$6 (2 x) + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [\frac{9}{2 x^{4}}]$

Step 2.2.3

Multiply $2$ by $6$ .

$12 x + \frac{d}{d x} [- 5 x] + \frac{d}{d x} [\frac{9}{2 x^{4}}]$

Step 2.3

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

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Step 2.3.1

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

$12 x - 5 \frac{d}{d x} [x] + \frac{d}{d x} [\frac{9}{2 x^{4}}]$

Step 2.3.2

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

$12 x - 5 \cdot 1 + \frac{d}{d x} [\frac{9}{2 x^{4}}]$

Step 2.3.3

Multiply $- 5$ by $1$ .

$12 x - 5 + \frac{d}{d x} [\frac{9}{2 x^{4}}]$

Step 2.4

Evaluate $\frac{d}{d x} [\frac{9}{2 x^{4}}]$ .

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Step 2.4.1

Since $\frac{9}{2}$ is constant with respect to $x$ , the derivative of $\frac{9}{2 x^{4}}$ with respect to $x$ is $\frac{9}{2} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$12 x - 5 + \frac{9}{2} \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 2.4.2

Rewrite $\frac{1}{x^{4}}$ as ${(x^{4})}^{- 1}$ .

$12 x - 5 + \frac{9}{2} \frac{d}{d x} [{(x^{4})}^{- 1}]$

Step 2.4.3

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 $g (x) = x^{4}$ .

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Step 2.4.3.1

To apply the Chain Rule, set $u$ as $x^{4}$ .

$12 x - 5 + \frac{9}{2} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}])$

Step 2.4.3.2

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

$12 x - 5 + \frac{9}{2} (- u^{- 2} \frac{d}{d x} [x^{4}])$

Step 2.4.3.3

Replace all occurrences of $u$ with $x^{4}$ .

$12 x - 5 + \frac{9}{2} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 2.4.4

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

$12 x - 5 + \frac{9}{2} (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 2.4.5

Multiply the exponents in ${(x^{4})}^{- 2}$ .

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Step 2.4.5.1

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

$12 x - 5 + \frac{9}{2} (- x^{4 \cdot - 2} (4 x^{3}))$

Step 2.4.5.2

Multiply $4$ by $- 2$ .

$12 x - 5 + \frac{9}{2} (- x^{- 8} (4 x^{3}))$

Step 2.4.6

Multiply $4$ by $- 1$ .

$12 x - 5 + \frac{9}{2} (- 4 x^{- 8} x^{3})$

Step 2.4.7

Multiply $x^{- 8}$ by $x^{3}$ by adding the exponents.

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Step 2.4.7.1

Move $x^{3}$ .

$12 x - 5 + \frac{9}{2} (- 4 (x^{3} x^{- 8}))$

Step 2.4.7.2

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

$12 x - 5 + \frac{9}{2} (- 4 x^{3 - 8})$

Step 2.4.7.3

Subtract $8$ from $3$ .

$12 x - 5 + \frac{9}{2} (- 4 x^{- 5})$

Step 2.4.8

Combine $- 4$ and $\frac{9}{2}$ .

$12 x - 5 + \frac{- 4 \cdot 9}{2} x^{- 5}$

Step 2.4.9

Multiply $- 4$ by $9$ .

$12 x - 5 + \frac{- 36}{2} x^{- 5}$

Step 2.4.10

Combine $\frac{- 36}{2}$ and $x^{- 5}$ .

$12 x - 5 + \frac{- 36 x^{- 5}}{2}$

Step 2.4.11

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

$12 x - 5 + \frac{- 36}{2 x^{5}}$

Step 2.4.12

Cancel the common factor of $- 36$ and $2$ .

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Step 2.4.12.1

Factor $2$ out of $- 36$ .

$12 x - 5 + \frac{2 \cdot - 18}{2 x^{5}}$

Step 2.4.12.2

Cancel the common factors.

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Step 2.4.12.2.1

Factor $2$ out of $2 x^{5}$ .

$12 x - 5 + \frac{2 \cdot - 18}{2 (x^{5})}$

Step 2.4.12.2.2

Cancel the common factor.

$12 x - 5 + \frac{2 \cdot - 18}{2 x^{5}}$

Step 2.4.12.2.3

Rewrite the expression.

$12 x - 5 + \frac{- 18}{x^{5}}$

Step 2.4.13

Move the negative in front of the fraction.

$12 x - 5 - \frac{18}{x^{5}}$

Step 2.5

Reorder terms.

$f'' (x) = 12 x - \frac{18}{x^{5}} - 5$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $12 x - \frac{18}{x^{5}} - 5$ .

$12 x - \frac{18}{x^{5}} - 5$