Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{1}{4} x^{- 3} - \frac{2}{3} - \frac{2}{3} x^{6} + \frac{1}{2} x^{4}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{4} x^{- 3}] + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$ .

$\frac{d}{d x} [\frac{1}{4} x^{- 3}] + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{1}{4} x^{- 3}]$ .

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Step 1.2.1

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{1}{4} x^{- 3}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x^{- 3}]$ .

$\frac{1}{4} \frac{d}{d x} [x^{- 3}] + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.2.2

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

$\frac{1}{4} (- 3 x^{- 4}) + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.2.3

Combine $- 3$ and $\frac{1}{4}$ .

$\frac{- 3}{4} x^{- 4} + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.2.4

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

$\frac{- 3 x^{- 4}}{4} + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.2.5

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

$\frac{- 3}{4 x^{4}} + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.2.6

Move the negative in front of the fraction.

$- \frac{3}{4 x^{4}} + \frac{d}{d x} [- \frac{2}{3}] + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.3

Since $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2}{3}$ with respect to $x$ is $0$ .

$- \frac{3}{4 x^{4}} + 0 + \frac{d}{d x} [- \frac{2}{3} x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4

Evaluate $\frac{d}{d x} [- \frac{2}{3} x^{6}]$ .

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Step 1.4.1

Since $- \frac{2}{3}$ is constant with respect to $x$ , the derivative of $- \frac{2}{3} x^{6}$ with respect to $x$ is $- \frac{2}{3} \frac{d}{d x} [x^{6}]$ .

$- \frac{3}{4 x^{4}} + 0 - \frac{2}{3} \frac{d}{d x} [x^{6}] + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 6$ .

$- \frac{3}{4 x^{4}} + 0 - \frac{2}{3} (6 x^{5}) + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.3

Multiply $6$ by $- 1$ .

$- \frac{3}{4 x^{4}} + 0 - 6 (\frac{2}{3}) x^{5} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.4

Combine $- 6$ and $\frac{2}{3}$ .

$- \frac{3}{4 x^{4}} + 0 + \frac{- 6 \cdot 2}{3} x^{5} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.5

Multiply $- 6$ by $2$ .

$- \frac{3}{4 x^{4}} + 0 + \frac{- 12}{3} x^{5} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.6

Combine $\frac{- 12}{3}$ and $x^{5}$ .

$- \frac{3}{4 x^{4}} + 0 + \frac{- 12 x^{5}}{3} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.7

Cancel the common factor of $- 12$ and $3$ .

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Step 1.4.7.1

Factor $3$ out of $- 12 x^{5}$ .

$- \frac{3}{4 x^{4}} + 0 + \frac{3 (- 4 x^{5})}{3} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.7.2

Cancel the common factors.

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Step 1.4.7.2.1

Factor $3$ out of $3$ .

$- \frac{3}{4 x^{4}} + 0 + \frac{3 (- 4 x^{5})}{3 (1)} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.7.2.2

Cancel the common factor.

$- \frac{3}{4 x^{4}} + 0 + \frac{3 (- 4 x^{5})}{3 \cdot 1} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.7.2.3

Rewrite the expression.

$- \frac{3}{4 x^{4}} + 0 + \frac{- 4 x^{5}}{1} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.4.7.2.4

Divide $- 4 x^{5}$ by $1$ .

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{d}{d x} [\frac{1}{2} x^{4}]$

Step 1.5

Evaluate $\frac{d}{d x} [\frac{1}{2} x^{4}]$ .

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Step 1.5.1

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

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{1}{2} \frac{d}{d x} [x^{4}]$

Step 1.5.2

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

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{1}{2} (4 x^{3})$

Step 1.5.3

Combine $4$ and $\frac{1}{2}$ .

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{4}{2} x^{3}$

Step 1.5.4

Combine $\frac{4}{2}$ and $x^{3}$ .

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{4 x^{3}}{2}$

Step 1.5.5

Cancel the common factor of $4$ and $2$ .

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Step 1.5.5.1

Factor $2$ out of $4 x^{3}$ .

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{2 (2 x^{3})}{2}$

Step 1.5.5.2

Cancel the common factors.

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Step 1.5.5.2.1

Factor $2$ out of $2$ .

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{2 (2 x^{3})}{2 (1)}$

Step 1.5.5.2.2

Cancel the common factor.

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{2 (2 x^{3})}{2 \cdot 1}$

Step 1.5.5.2.3

Rewrite the expression.

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + \frac{2 x^{3}}{1}$

Step 1.5.5.2.4

Divide $2 x^{3}$ by $1$ .

$- \frac{3}{4 x^{4}} + 0 - 4 x^{5} + 2 x^{3}$

Step 1.6

Simplify.

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Step 1.6.1

Add $- \frac{3}{4 x^{4}}$ and $0$ .

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

Step 1.6.2

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $5$ by $- 4$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $3$ by $2$ .

$- 20 x^{4} + 6 x^{2} + \frac{d}{d x} [- \frac{3}{4 x^{4}}]$

Step 2.4

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

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Step 2.4.1

Since $- \frac{3}{4}$ is constant with respect to $x$ , the derivative of $- \frac{3}{4 x^{4}}$ with respect to $x$ is $- \frac{3}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} \frac{d}{d x} [\frac{1}{x^{4}}]$

Step 2.4.2

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

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} \frac{d}{d x} [{(x^{4})}^{- 1}]$

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 $x^{4}$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}])$

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 $n = - 1$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- u^{- 2} \frac{d}{d x} [x^{4}])$

Step 2.4.3.3

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

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 2.4.4

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

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- {(x^{4})}^{- 2} (4 x^{3}))$

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, ${(a^{m})}^{n} = a^{m n}$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- x^{4 \cdot - 2} (4 x^{3}))$

Step 2.4.5.2

Multiply $4$ by $- 2$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- x^{- 8} (4 x^{3}))$

Step 2.4.6

Multiply $4$ by $- 1$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- 4 x^{- 8} x^{3})$

Step 2.4.7

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

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Step 2.4.7.1

Move $x^{3}$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- 4 (x^{3} x^{- 8}))$

Step 2.4.7.2

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

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- 4 x^{3 - 8})$

Step 2.4.7.3

Subtract $8$ from $3$ .

$- 20 x^{4} + 6 x^{2} - \frac{3}{4} (- 4 x^{- 5})$

Step 2.4.8

Multiply $- 4$ by $- 1$ .

$- 20 x^{4} + 6 x^{2} + 4 (\frac{3}{4}) x^{- 5}$

Step 2.4.9

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

$- 20 x^{4} + 6 x^{2} + \frac{4 \cdot 3}{4} x^{- 5}$

Step 2.4.10

Multiply $4$ by $3$ .

$- 20 x^{4} + 6 x^{2} + \frac{12}{4} x^{- 5}$

Step 2.4.11

Combine $\frac{12}{4}$ and $x^{- 5}$ .

$- 20 x^{4} + 6 x^{2} + \frac{12 x^{- 5}}{4}$

Step 2.4.12

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

$- 20 x^{4} + 6 x^{2} + \frac{12}{4 x^{5}}$

Step 2.4.13

Cancel the common factor of $12$ and $4$ .

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Step 2.4.13.1

Factor $4$ out of $12$ .

$- 20 x^{4} + 6 x^{2} + \frac{4 \cdot 3}{4 x^{5}}$

Step 2.4.13.2

Cancel the common factors.

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Step 2.4.13.2.1

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

$- 20 x^{4} + 6 x^{2} + \frac{4 \cdot 3}{4 (x^{5})}$

Step 2.4.13.2.2

Cancel the common factor.

$- 20 x^{4} + 6 x^{2} + \frac{4 \cdot 3}{4 x^{5}}$

Step 2.4.13.2.3

Rewrite the expression.

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

Step 3

The second derivative of $f (x)$ with respect to $x$ is $- 20 x^{4} + 6 x^{2} + \frac{3}{x^{5}}$ .

$- 20 x^{4} + 6 x^{2} + \frac{3}{x^{5}}$