Calculus Examples

$y = - \frac{1}{24} x^{- 4} - \frac{3}{20} x - x^{3} + \frac{5}{18}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $- \frac{1}{24} x^{- 4} - \frac{3}{20} x - x^{3} + \frac{5}{18}$ with respect to $x$ is $\frac{d}{d x} [- \frac{1}{24} x^{- 4}] + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$ .

$\frac{d}{d x} [- \frac{1}{24} x^{- 4}] + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2

Evaluate $\frac{d}{d x} [- \frac{1}{24} x^{- 4}]$ .

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Step 1.2.1

Since $- \frac{1}{24}$ is constant with respect to $x$ , the derivative of $- \frac{1}{24} x^{- 4}$ with respect to $x$ is $- \frac{1}{24} \frac{d}{d x} [x^{- 4}]$ .

$- \frac{1}{24} \frac{d}{d x} [x^{- 4}] + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.2

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

$- \frac{1}{24} (- 4 x^{- 5}) + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.3

Multiply $- 4$ by $- 1$ .

$4 (\frac{1}{24}) x^{- 5} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.4

Combine $4$ and $\frac{1}{24}$ .

$\frac{4}{24} x^{- 5} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.5

Combine $\frac{4}{24}$ and $x^{- 5}$ .

$\frac{4 x^{- 5}}{24} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.6

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

$\frac{4}{24 x^{5}} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.7

Cancel the common factor of $4$ and $24$ .

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Step 1.2.7.1

Factor $4$ out of $4$ .

$\frac{4 (1)}{24 x^{5}} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.7.2

Cancel the common factors.

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Step 1.2.7.2.1

Factor $4$ out of $24 x^{5}$ .

$\frac{4 (1)}{4 (6 x^{5})} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.7.2.2

Cancel the common factor.

$\frac{4 \cdot 1}{4 (6 x^{5})} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.2.7.2.3

Rewrite the expression.

$\frac{1}{6 x^{5}} + \frac{d}{d x} [- \frac{3}{20} x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.3

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

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Step 1.3.1

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

$\frac{1}{6 x^{5}} - \frac{3}{20} \frac{d}{d x} [x] + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.3.2

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

$\frac{1}{6 x^{5}} - \frac{3}{20} \cdot 1 + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.3.3

Multiply $- 1$ by $1$ .

$\frac{1}{6 x^{5}} - \frac{3}{20} + \frac{d}{d x} [- x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.4

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

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Step 1.4.1

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

$\frac{1}{6 x^{5}} - \frac{3}{20} - \frac{d}{d x} [x^{3}] + \frac{d}{d x} [\frac{5}{18}]$

Step 1.4.2

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

$\frac{1}{6 x^{5}} - \frac{3}{20} - (3 x^{2}) + \frac{d}{d x} [\frac{5}{18}]$

Step 1.4.3

Multiply $3$ by $- 1$ .

$\frac{1}{6 x^{5}} - \frac{3}{20} - 3 x^{2} + \frac{d}{d x} [\frac{5}{18}]$

Step 1.5

Since $\frac{5}{18}$ is constant with respect to $x$ , the derivative of $\frac{5}{18}$ with respect to $x$ is $0$ .

$\frac{1}{6 x^{5}} - \frac{3}{20} - 3 x^{2} + 0$

Step 1.6

Simplify.

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Step 1.6.1

Add $\frac{1}{6 x^{5}} - \frac{3}{20} - 3 x^{2}$ and $0$ .

$\frac{1}{6 x^{5}} - \frac{3}{20} - 3 x^{2}$

Step 1.6.2

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $- 3 x^{2} + \frac{1}{6 x^{5}} - \frac{3}{20}$ with respect to $x$ is $\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [\frac{1}{6 x^{5}}] + \frac{d}{d x} [- \frac{3}{20}]$ .

$\frac{d}{d x} [- 3 x^{2}] + \frac{d}{d x} [\frac{1}{6 x^{5}}] + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.2

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

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Step 2.2.1

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

$- 3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [\frac{1}{6 x^{5}}] + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.2.2

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

$- 3 (2 x) + \frac{d}{d x} [\frac{1}{6 x^{5}}] + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.2.3

Multiply $2$ by $- 3$ .

$- 6 x + \frac{d}{d x} [\frac{1}{6 x^{5}}] + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3

Evaluate $\frac{d}{d x} [\frac{1}{6 x^{5}}]$ .

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Step 2.3.1

Since $\frac{1}{6}$ is constant with respect to $x$ , the derivative of $\frac{1}{6 x^{5}}$ with respect to $x$ is $\frac{1}{6} \frac{d}{d x} [\frac{1}{x^{5}}]$ .

$- 6 x + \frac{1}{6} \frac{d}{d x} [\frac{1}{x^{5}}] + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.2

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

$- 6 x + \frac{1}{6} \frac{d}{d x} [{(x^{5})}^{- 1}] + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.3

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 2.3.3.1

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

$- 6 x + \frac{1}{6} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.3.2

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

$- 6 x + \frac{1}{6} (- u^{- 2} \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.3.3

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

$- 6 x + \frac{1}{6} (- {(x^{5})}^{- 2} \frac{d}{d x} [x^{5}]) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.4

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

$- 6 x + \frac{1}{6} (- {(x^{5})}^{- 2} (5 x^{4})) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.5

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

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Step 2.3.5.1

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

$- 6 x + \frac{1}{6} (- x^{5 \cdot - 2} (5 x^{4})) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.5.2

Multiply $5$ by $- 2$ .

$- 6 x + \frac{1}{6} (- x^{- 10} (5 x^{4})) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.6

Multiply $5$ by $- 1$ .

$- 6 x + \frac{1}{6} (- 5 x^{- 10} x^{4}) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.7

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

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Step 2.3.7.1

Move $x^{4}$ .

$- 6 x + \frac{1}{6} (- 5 (x^{4} x^{- 10})) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.7.2

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

$- 6 x + \frac{1}{6} (- 5 x^{4 - 10}) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.7.3

Subtract $10$ from $4$ .

$- 6 x + \frac{1}{6} (- 5 x^{- 6}) + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.8

Combine $- 5$ and $\frac{1}{6}$ .

$- 6 x + \frac{- 5}{6} x^{- 6} + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.9

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

$- 6 x + \frac{- 5 x^{- 6}}{6} + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.10

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

$- 6 x + \frac{- 5}{6 x^{6}} + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.3.11

Move the negative in front of the fraction.

$- 6 x - \frac{5}{6 x^{6}} + \frac{d}{d x} [- \frac{3}{20}]$

Step 2.4

Differentiate using the Constant Rule.

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Step 2.4.1

Since $- \frac{3}{20}$ is constant with respect to $x$ , the derivative of $- \frac{3}{20}$ with respect to $x$ is $0$ .

$- 6 x - \frac{5}{6 x^{6}} + 0$

Step 2.4.2

Add $- 6 x - \frac{5}{6 x^{6}}$ and $0$ .

$f'' (x) = - 6 x - \frac{5}{6 x^{6}}$

Step 3

Find the third derivative.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

Multiply $- 6$ by $1$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [- \frac{5}{6 x^{6}}]$ .

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Step 3.3.1

Since $- \frac{5}{6}$ is constant with respect to $x$ , the derivative of $- \frac{5}{6 x^{6}}$ with respect to $x$ is $- \frac{5}{6} \frac{d}{d x} [\frac{1}{x^{6}}]$ .

$- 6 - \frac{5}{6} \frac{d}{d x} [\frac{1}{x^{6}}]$

Step 3.3.2

Rewrite $\frac{1}{x^{6}}$ as ${(x^{6})}^{- 1}$ .

$- 6 - \frac{5}{6} \frac{d}{d x} [{(x^{6})}^{- 1}]$

Step 3.3.3

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 $g (x) = x^{6}$ .

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Step 3.3.3.1

To apply the Chain Rule, set $u$ as $x^{6}$ .

$- 6 - \frac{5}{6} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{6}])$

Step 3.3.3.2

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

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

Step 3.3.3.3

Replace all occurrences of $u$ with $x^{6}$ .

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

Step 3.3.4

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

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

Step 3.3.5

Multiply the exponents in ${(x^{6})}^{- 2}$ .

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Step 3.3.5.1

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

$- 6 - \frac{5}{6} (- x^{6 \cdot - 2} (6 x^{5}))$

Step 3.3.5.2

Multiply $6$ by $- 2$ .

$- 6 - \frac{5}{6} (- x^{- 12} (6 x^{5}))$

Step 3.3.6

Multiply $6$ by $- 1$ .

$- 6 - \frac{5}{6} (- 6 x^{- 12} x^{5})$

Step 3.3.7

Multiply $x^{- 12}$ by $x^{5}$ by adding the exponents.

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Step 3.3.7.1

Move $x^{5}$ .

$- 6 - \frac{5}{6} (- 6 (x^{5} x^{- 12}))$

Step 3.3.7.2

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

$- 6 - \frac{5}{6} (- 6 x^{5 - 12})$

Step 3.3.7.3

Subtract $12$ from $5$ .

$- 6 - \frac{5}{6} (- 6 x^{- 7})$

Step 3.3.8

Multiply $- 6$ by $- 1$ .

$- 6 + 6 (\frac{5}{6}) x^{- 7}$

Step 3.3.9

Combine $6$ and $\frac{5}{6}$ .

$- 6 + \frac{6 \cdot 5}{6} x^{- 7}$

Step 3.3.10

Multiply $6$ by $5$ .

$- 6 + \frac{30}{6} x^{- 7}$

Step 3.3.11

Combine $\frac{30}{6}$ and $x^{- 7}$ .

$- 6 + \frac{30 x^{- 7}}{6}$

Step 3.3.12

Move $x^{- 7}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 6 + \frac{30}{6 x^{7}}$

Step 3.3.13

Cancel the common factor of $30$ and $6$ .

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Step 3.3.13.1

Factor $6$ out of $30$ .

$- 6 + \frac{6 \cdot 5}{6 x^{7}}$

Step 3.3.13.2

Cancel the common factors.

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Step 3.3.13.2.1

Factor $6$ out of $6 x^{7}$ .

$- 6 + \frac{6 \cdot 5}{6 (x^{7})}$

Step 3.3.13.2.2

Cancel the common factor.

$- 6 + \frac{6 \cdot 5}{6 x^{7}}$

Step 3.3.13.2.3

Rewrite the expression.

$- 6 + \frac{5}{x^{7}}$

Step 3.4

Reorder terms.

$f''' (x) = \frac{5}{x^{7}} - 6$