Calculus Examples

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

Step 1

Find the first derivative.

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Step 1.1

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

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

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

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

Step 1.2.2

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

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

Step 1.2.3

Multiply $2$ by $3$ .

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

Multiply $- 3$ by $2$ .

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

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.3.7

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

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Step 1.3.7.1

Factor $3$ out of $- 6$ .

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

Step 1.3.7.2

Cancel the common factors.

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Step 1.3.7.2.1

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

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

Step 1.3.7.2.2

Cancel the common factor.

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

Step 1.3.7.2.3

Rewrite the expression.

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

Step 1.3.8

Move the negative in front of the fraction.

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.4.3

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

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

Step 1.4.4

Multiply $3$ by $2$ .

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

Step 1.4.5

Combine $\frac{6}{3}$ and $x^{2}$ .

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

Step 1.4.6

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

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Step 1.4.6.1

Factor $3$ out of $6 x^{2}$ .

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

Step 1.4.6.2

Cancel the common factors.

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Step 1.4.6.2.1

Factor $3$ out of $3$ .

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

Step 1.4.6.2.2

Cancel the common factor.

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

Step 1.4.6.2.3

Rewrite the expression.

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

Step 1.4.6.2.4

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

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

Step 1.5

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

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Step 1.5.1

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

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

Step 1.5.2

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

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

Step 1.5.3

Multiply $- 1$ by $- 1$ .

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

Step 1.5.4

Multiply $x^{- 2}$ by $1$ .

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

Step 1.6

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

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

Step 1.7

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Multiply $2$ by $2$ .

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $6$ by $1$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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 2.4.2.1

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

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

Step 2.4.2.2

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

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

Step 2.4.2.3

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

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

Step 2.4.3

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

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

Step 2.4.4

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

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Step 2.4.4.1

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

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

Step 2.4.4.2

Multiply $2$ by $- 2$ .

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

Step 2.4.5

Multiply $2$ by $- 1$ .

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

Step 2.4.6

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

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

Step 2.4.7

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

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

Step 2.4.8

Subtract $4$ from $1$ .

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

Step 2.5

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

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Step 2.5.1

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

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

Step 2.5.2

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

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

Step 2.5.3

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 2.5.3.1

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

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

Step 2.5.3.2

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

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

Step 2.5.3.3

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

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

Step 2.5.4

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

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

Step 2.5.5

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

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Step 2.5.5.1

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

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

Step 2.5.5.2

Multiply $4$ by $- 2$ .

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

Step 2.5.6

Multiply $4$ by $- 1$ .

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

Step 2.5.7

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

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Step 2.5.7.1

Move $x^{3}$ .

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

Step 2.5.7.2

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

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

Step 2.5.7.3

Subtract $8$ from $3$ .

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

Step 2.5.8

Multiply $- 4$ by $- 2$ .

$4 x + 6 - 2 x^{- 3} + 8 x^{- 5}$

Step 2.6

Simplify.

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Step 2.6.1

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

$4 x + 6 - 2 \frac{1}{x^{3}} + 8 x^{- 5}$

Step 2.6.2

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

$4 x + 6 - 2 \frac{1}{x^{3}} + 8 \frac{1}{x^{5}}$

Step 2.6.3

Combine terms.

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Step 2.6.3.1

Combine $- 2$ and $\frac{1}{x^{3}}$ .

$4 x + 6 + \frac{- 2}{x^{3}} + 8 \frac{1}{x^{5}}$

Step 2.6.3.2

Move the negative in front of the fraction.

$4 x + 6 - \frac{2}{x^{3}} + 8 \frac{1}{x^{5}}$

Step 2.6.3.3

Combine $8$ and $\frac{1}{x^{5}}$ .

$4 x + 6 - \frac{2}{x^{3}} + \frac{8}{x^{5}}$

Step 2.6.4

Reorder terms.

$f'' (x) = 4 x - \frac{2}{x^{3}} + \frac{8}{x^{5}} + 6$

Step 3

The second derivative of $f (x)$ with respect to $x$ is $4 x - \frac{2}{x^{3}} + \frac{8}{x^{5}} + 6$ .

$4 x - \frac{2}{x^{3}} + \frac{8}{x^{5}} + 6$