Calculus Examples

$y = 3 x^{- 2} + 4 x^{- 1} + x$

Step 1

Find the first derivative.

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $- 2$ by $3$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $- 1$ by $4$ .

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

Step 1.4

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

$- 6 x^{- 3} - 4 x^{- 2} + 1$

Step 1.5

Simplify.

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Step 1.5.1

Reorder terms.

$- 4 x^{- 2} - 6 x^{- 3} + 1$

Step 1.5.2

Simplify each term.

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Step 1.5.2.1

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

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

Step 1.5.2.2

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

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

Step 1.5.2.3

Move the negative in front of the fraction.

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

Step 1.5.2.4

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

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

Step 1.5.2.5

Combine $- 6$ and $\frac{1}{x^{3}}$ .

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

Step 1.5.2.6

Move the negative in front of the fraction.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Rewrite $\frac{1}{x^{2}}$ as ${(x^{2})}^{- 1}$ .

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

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 $g (x) = x^{2}$ .

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Step 2.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{2}$ .

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

Step 2.2.3.2

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

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

Step 2.2.3.3

Replace all occurrences of $u_{1}$ with $x^{2}$ .

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

Step 2.2.4

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

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

Step 2.2.5

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

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Step 2.2.5.1

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

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

Step 2.2.5.2

Multiply $2$ by $- 2$ .

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

Step 2.2.6

Multiply $2$ by $- 1$ .

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

Step 2.2.7

Raise $x$ to the power of $1$ .

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

Step 2.2.8

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

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

Step 2.2.9

Subtract $4$ from $1$ .

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

Step 2.2.10

Multiply $- 2$ by $- 4$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

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

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 $g (x) = x^{3}$ .

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Step 2.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{3}$ .

$8 x^{- 3} - 6 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [1]$

Step 2.3.3.2

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

$8 x^{- 3} - 6 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [1]$

Step 2.3.3.3

Replace all occurrences of $u_{2}$ with $x^{3}$ .

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

Step 2.3.4

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

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

Step 2.3.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 2.3.5.1

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

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

Step 2.3.5.2

Multiply $3$ by $- 2$ .

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

Step 2.3.6

Multiply $3$ by $- 1$ .

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

Step 2.3.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 2.3.7.1

Move $x^{2}$ .

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

Step 2.3.7.2

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

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

Step 2.3.7.3

Subtract $6$ from $2$ .

$8 x^{- 3} - 6 (- 3 x^{- 4}) + \frac{d}{d x} [1]$

Step 2.3.8

Multiply $- 3$ by $- 6$ .

$8 x^{- 3} + 18 x^{- 4} + \frac{d}{d x} [1]$

Step 2.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$8 x^{- 3} + 18 x^{- 4} + 0$

Step 2.5

Simplify.

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Step 2.5.1

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

$8 \frac{1}{x^{3}} + 18 x^{- 4} + 0$

Step 2.5.2

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

$8 \frac{1}{x^{3}} + 18 \frac{1}{x^{4}} + 0$

Step 2.5.3

Combine terms.

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Step 2.5.3.1

Combine $8$ and $\frac{1}{x^{3}}$ .

$\frac{8}{x^{3}} + 18 \frac{1}{x^{4}} + 0$

Step 2.5.3.2

Combine $18$ and $\frac{1}{x^{4}}$ .

$\frac{8}{x^{3}} + \frac{18}{x^{4}} + 0$

Step 2.5.3.3

Add $\frac{8}{x^{3}} + \frac{18}{x^{4}}$ and $0$ .

$f'' (x) = \frac{8}{x^{3}} + \frac{18}{x^{4}}$

Step 3

Find the third derivative.

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Step 3.1

By the Sum Rule, the derivative of $\frac{8}{x^{3}} + \frac{18}{x^{4}}$ with respect to $x$ is $\frac{d}{d x} [\frac{8}{x^{3}}] + \frac{d}{d x} [\frac{18}{x^{4}}]$ .

$\frac{d}{d x} [\frac{8}{x^{3}}] + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2

Evaluate $\frac{d}{d x} [\frac{8}{x^{3}}]$ .

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Step 3.2.1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8}{x^{3}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{1}{x^{3}}]$ .

$8 \frac{d}{d x} [\frac{1}{x^{3}}] + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.2

Rewrite $\frac{1}{x^{3}}$ as ${(x^{3})}^{- 1}$ .

$8 \frac{d}{d x} [{(x^{3})}^{- 1}] + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.3

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 $g (x) = x^{3}$ .

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Step 3.2.3.1

To apply the Chain Rule, set $u_{1}$ as $x^{3}$ .

$8 (\frac{d}{d u_{1}} [{u_{1}}^{- 1}] \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.3.2

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

$8 (- {u_{1}}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.3.3

Replace all occurrences of $u_{1}$ with $x^{3}$ .

$8 (- {(x^{3})}^{- 2} \frac{d}{d x} [x^{3}]) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.4

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

$8 (- {(x^{3})}^{- 2} (3 x^{2})) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.5

Multiply the exponents in ${(x^{3})}^{- 2}$ .

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Step 3.2.5.1

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

$8 (- x^{3 \cdot - 2} (3 x^{2})) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.5.2

Multiply $3$ by $- 2$ .

$8 (- x^{- 6} (3 x^{2})) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.6

Multiply $3$ by $- 1$ .

$8 (- 3 x^{- 6} x^{2}) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.7

Multiply $x^{- 6}$ by $x^{2}$ by adding the exponents.

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Step 3.2.7.1

Move $x^{2}$ .

$8 (- 3 (x^{2} x^{- 6})) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.7.2

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

$8 (- 3 x^{2 - 6}) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.7.3

Subtract $6$ from $2$ .

$8 (- 3 x^{- 4}) + \frac{d}{d x} [\frac{18}{x^{4}}]$

Step 3.2.8

Multiply $- 3$ by $8$ .

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

Step 3.3

Evaluate $\frac{d}{d x} [\frac{18}{x^{4}}]$ .

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Step 3.3.1

Since $18$ is constant with respect to $x$ , the derivative of $\frac{18}{x^{4}}$ with respect to $x$ is $18 \frac{d}{d x} [\frac{1}{x^{4}}]$ .

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

Step 3.3.2

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

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

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

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Step 3.3.3.1

To apply the Chain Rule, set $u_{2}$ as $x^{4}$ .

$- 24 x^{- 4} + 18 (\frac{d}{d u_{2}} [{u_{2}}^{- 1}] \frac{d}{d x} [x^{4}])$

Step 3.3.3.2

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

$- 24 x^{- 4} + 18 (- {u_{2}}^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.3.3.3

Replace all occurrences of $u_{2}$ with $x^{4}$ .

$- 24 x^{- 4} + 18 (- {(x^{4})}^{- 2} \frac{d}{d x} [x^{4}])$

Step 3.3.4

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

$- 24 x^{- 4} + 18 (- {(x^{4})}^{- 2} (4 x^{3}))$

Step 3.3.5

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

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Step 3.3.5.1

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

$- 24 x^{- 4} + 18 (- x^{4 \cdot - 2} (4 x^{3}))$

Step 3.3.5.2

Multiply $4$ by $- 2$ .

$- 24 x^{- 4} + 18 (- x^{- 8} (4 x^{3}))$

Step 3.3.6

Multiply $4$ by $- 1$ .

$- 24 x^{- 4} + 18 (- 4 x^{- 8} x^{3})$

Step 3.3.7

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

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Step 3.3.7.1

Move $x^{3}$ .

$- 24 x^{- 4} + 18 (- 4 (x^{3} x^{- 8}))$

Step 3.3.7.2

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

$- 24 x^{- 4} + 18 (- 4 x^{3 - 8})$

Step 3.3.7.3

Subtract $8$ from $3$ .

$- 24 x^{- 4} + 18 (- 4 x^{- 5})$

Step 3.3.8

Multiply $- 4$ by $18$ .

$- 24 x^{- 4} - 72 x^{- 5}$

Step 3.4

Simplify.

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Step 3.4.1

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

$- 24 \frac{1}{x^{4}} - 72 x^{- 5}$

Step 3.4.2

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

$- 24 \frac{1}{x^{4}} - 72 \frac{1}{x^{5}}$

Step 3.4.3

Combine terms.

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Step 3.4.3.1

Combine $- 24$ and $\frac{1}{x^{4}}$ .

$\frac{- 24}{x^{4}} - 72 \frac{1}{x^{5}}$

Step 3.4.3.2

Move the negative in front of the fraction.

$- \frac{24}{x^{4}} - 72 \frac{1}{x^{5}}$

Step 3.4.3.3

Combine $- 72$ and $\frac{1}{x^{5}}$ .

$- \frac{24}{x^{4}} + \frac{- 72}{x^{5}}$

Step 3.4.3.4

Move the negative in front of the fraction.

$f''' (x) = - \frac{24}{x^{4}} - \frac{72}{x^{5}}$