Calculus Examples

$y = \frac{5}{8} x + \frac{2}{9} - \frac{1}{6} x^{- 3}$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $\frac{5}{8} x + \frac{2}{9} - \frac{1}{6} x^{- 3}$ with respect to $x$ is $\frac{d}{d x} [\frac{5}{8} x] + \frac{d}{d x} [\frac{2}{9}] + \frac{d}{d x} [- \frac{1}{6} x^{- 3}]$ .

$\frac{d}{d x} [\frac{5}{8} x] + \frac{d}{d x} [\frac{2}{9}] + \frac{d}{d x} [- \frac{1}{6} x^{- 3}]$

Step 1.2

Evaluate $\frac{d}{d x} [\frac{5}{8} x]$ .

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Step 1.2.1

Since $\frac{5}{8}$ is constant with respect to $x$ , the derivative of $\frac{5}{8} x$ with respect to $x$ is $\frac{5}{8} \frac{d}{d x} [x]$ .

$\frac{5}{8} \frac{d}{d x} [x] + \frac{d}{d x} [\frac{2}{9}] + \frac{d}{d x} [- \frac{1}{6} x^{- 3}]$

Step 1.2.2

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

$\frac{5}{8} \cdot 1 + \frac{d}{d x} [\frac{2}{9}] + \frac{d}{d x} [- \frac{1}{6} x^{- 3}]$

Step 1.2.3

Multiply $\frac{5}{8}$ by $1$ .

$\frac{5}{8} + \frac{d}{d x} [\frac{2}{9}] + \frac{d}{d x} [- \frac{1}{6} x^{- 3}]$

Step 1.3

Since $\frac{2}{9}$ is constant with respect to $x$ , the derivative of $\frac{2}{9}$ with respect to $x$ is $0$ .

$\frac{5}{8} + 0 + \frac{d}{d x} [- \frac{1}{6} x^{- 3}]$

Step 1.4

Evaluate $\frac{d}{d x} [- \frac{1}{6} x^{- 3}]$ .

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Step 1.4.1

Since $- \frac{1}{6}$ is constant with respect to $x$ , the derivative of $- \frac{1}{6} x^{- 3}$ with respect to $x$ is $- \frac{1}{6} \frac{d}{d x} [x^{- 3}]$ .

$\frac{5}{8} + 0 - \frac{1}{6} \frac{d}{d x} [x^{- 3}]$

Step 1.4.2

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

$\frac{5}{8} + 0 - \frac{1}{6} (- 3 x^{- 4})$

Step 1.4.3

Multiply $- 3$ by $- 1$ .

$\frac{5}{8} + 0 + 3 (\frac{1}{6}) x^{- 4}$

Step 1.4.4

Combine $3$ and $\frac{1}{6}$ .

$\frac{5}{8} + 0 + \frac{3}{6} x^{- 4}$

Step 1.4.5

Combine $\frac{3}{6}$ and $x^{- 4}$ .

$\frac{5}{8} + 0 + \frac{3 x^{- 4}}{6}$

Step 1.4.6

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

$\frac{5}{8} + 0 + \frac{3}{6 x^{4}}$

Step 1.4.7

Cancel the common factor of $3$ and $6$ .

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Step 1.4.7.1

Factor $3$ out of $3$ .

$\frac{5}{8} + 0 + \frac{3 (1)}{6 x^{4}}$

Step 1.4.7.2

Cancel the common factors.

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Step 1.4.7.2.1

Factor $3$ out of $6 x^{4}$ .

$\frac{5}{8} + 0 + \frac{3 (1)}{3 (2 x^{4})}$

Step 1.4.7.2.2

Cancel the common factor.

$\frac{5}{8} + 0 + \frac{3 \cdot 1}{3 (2 x^{4})}$

Step 1.4.7.2.3

Rewrite the expression.

$\frac{5}{8} + 0 + \frac{1}{2 x^{4}}$

Step 1.5

Simplify.

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Step 1.5.1

Add $\frac{5}{8}$ and $0$ .

$\frac{5}{8} + \frac{1}{2 x^{4}}$

Step 1.5.2

Reorder terms.

$f' (x) = \frac{1}{2 x^{4}} + \frac{5}{8}$

Step 2

Find the second derivative.

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Step 2.1

By the Sum Rule, the derivative of $\frac{1}{2 x^{4}} + \frac{5}{8}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{2 x^{4}}] + \frac{d}{d x} [\frac{5}{8}]$ .

$\frac{d}{d x} [\frac{1}{2 x^{4}}] + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2

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

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Step 2.2.1

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

$\frac{1}{2} \frac{d}{d x} [\frac{1}{x^{4}}] + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.2

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

$\frac{1}{2} \frac{d}{d x} [{(x^{4})}^{- 1}] + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.3

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 2.2.3.1

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

$\frac{1}{2} (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.3.2

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

$\frac{1}{2} (- u^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.3.3

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

$\frac{1}{2} (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.4

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

$\frac{1}{2} (- {(x^{4})}^{- 2} (4 x^{3})) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.5

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

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Step 2.2.5.1

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

$\frac{1}{2} (- x^{4 \cdot - 2} (4 x^{3})) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.5.2

Multiply $4$ by $- 2$ .

$\frac{1}{2} (- x^{- 8} (4 x^{3})) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.6

Multiply $4$ by $- 1$ .

$\frac{1}{2} (- 4 x^{- 8} x^{3}) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.7

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

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Step 2.2.7.1

Move $x^{3}$ .

$\frac{1}{2} (- 4 (x^{3} x^{- 8})) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.7.2

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

$\frac{1}{2} (- 4 x^{3 - 8}) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.7.3

Subtract $8$ from $3$ .

$\frac{1}{2} (- 4 x^{- 5}) + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.8

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

$\frac{- 4}{2} x^{- 5} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.9

Combine $\frac{- 4}{2}$ and $x^{- 5}$ .

$\frac{- 4 x^{- 5}}{2} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.10

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

$\frac{- 4}{2 x^{5}} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.11

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

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Step 2.2.11.1

Factor $2$ out of $- 4$ .

$\frac{2 \cdot - 2}{2 x^{5}} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.11.2

Cancel the common factors.

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Step 2.2.11.2.1

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

$\frac{2 \cdot - 2}{2 (x^{5})} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.11.2.2

Cancel the common factor.

$\frac{2 \cdot - 2}{2 x^{5}} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.11.2.3

Rewrite the expression.

$\frac{- 2}{x^{5}} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.2.12

Move the negative in front of the fraction.

$- \frac{2}{x^{5}} + \frac{d}{d x} [\frac{5}{8}]$

Step 2.3

Differentiate using the Constant Rule.

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Step 2.3.1

Since $\frac{5}{8}$ is constant with respect to $x$ , the derivative of $\frac{5}{8}$ with respect to $x$ is $0$ .

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

Step 2.3.2

Add $- \frac{2}{x^{5}}$ and $0$ .

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

Step 3

Find the third derivative.

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Step 3.1

Since $- 2$ is constant with respect to $x$ , the derivative of $- \frac{2}{x^{5}}$ with respect to $x$ is $- 2 \frac{d}{d x} [\frac{1}{x^{5}}]$ .

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

Step 3.2

Apply basic rules of exponents.

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Step 3.2.1

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

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

Step 3.2.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 3.2.2.1

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

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

Step 3.2.2.2

Multiply $5$ by $- 1$ .

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

Step 3.3

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

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

Step 3.4

Multiply $- 5$ by $- 2$ .

$10 x^{- 6}$

Step 3.5

Simplify.

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Step 3.5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$10 \frac{1}{x^{6}}$

Step 3.5.2

Combine $10$ and $\frac{1}{x^{6}}$ .

$f''' (x) = \frac{10}{x^{6}}$