Calculus Examples

$f (x) = x + \frac{1}{x} + \sqrt{x}$

Step 1

Find the first derivative.

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Step 1.1

Differentiate.

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Step 1.1.1

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

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\sqrt{x}]$

Step 1.1.2

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

$1 + \frac{d}{d x} [\frac{1}{x}] + \frac{d}{d x} [\sqrt{x}]$

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.3

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

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Step 1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

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

Step 1.3.2

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

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

Step 1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1 - x^{- 2} + \frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 1.3.4

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

$1 - x^{- 2} + \frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 1.3.5

Combine the numerators over the common denominator.

$1 - x^{- 2} + \frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 1.3.6

Simplify the numerator.

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Step 1.3.6.1

Multiply $- 1$ by $2$ .

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

Step 1.3.6.2

Subtract $2$ from $1$ .

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

Step 1.3.7

Move the negative in front of the fraction.

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

Step 1.4

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

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

Step 1.5

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

$1 - \frac{1}{x^{2}} + \frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$

Step 1.6

Simplify.

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Step 1.6.1

Multiply $\frac{1}{2}$ by $\frac{1}{x^{\frac{1}{2}}}$ .

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

Step 1.6.2

Reorder terms.

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

Step 2

Find the second derivative.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = - 1$ and $g (x) = \frac{1}{x^{2}}$ .

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

Step 2.2.2

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

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

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 $x^{2}$ .

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

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 $n = - 1$ .

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

Step 2.2.3.3

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

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

Step 2.2.4

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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Step 2.2.6.1

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

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

Step 2.2.6.2

Multiply $2$ by $- 2$ .

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

Step 2.2.7

Multiply $2$ by $- 1$ .

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

Step 2.2.8

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

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

Step 2.2.9

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

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

Step 2.2.10

Subtract $4$ from $1$ .

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

Step 2.2.11

Multiply $- 2$ by $- 1$ .

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

Step 2.2.12

Multiply $\frac{1}{x^{2}}$ by $0$ .

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

Step 2.2.13

Add $2 x^{- 3}$ and $0$ .

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

Step 2.3

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

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

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

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 $g (x) = x^{\frac{1}{2}}$ .

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Step 2.4.3.1

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

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

Step 2.4.3.2

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

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

Step 2.4.3.3

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

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

Step 2.4.4

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

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

Step 2.4.5

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 2}$ .

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Step 2.4.5.1

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

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

Step 2.4.5.2

Cancel the common factor of $2$ .

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Step 2.4.5.2.1

Factor $2$ out of $- 2$ .

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

Step 2.4.5.2.2

Cancel the common factor.

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

Step 2.4.5.2.3

Rewrite the expression.

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

Step 2.4.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4.7

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

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

Step 2.4.8

Combine the numerators over the common denominator.

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

Step 2.4.9

Simplify the numerator.

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Step 2.4.9.1

Multiply $- 1$ by $2$ .

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

Step 2.4.9.2

Subtract $2$ from $1$ .

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

Step 2.4.10

Move the negative in front of the fraction.

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

Step 2.4.11

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

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

Step 2.4.12

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

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

Step 2.4.13

Multiply $x^{- \frac{1}{2}}$ by $x^{- 1}$ by adding the exponents.

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Step 2.4.13.1

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

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

Step 2.4.13.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.4.13.3

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

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

Step 2.4.13.4

Combine the numerators over the common denominator.

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

Step 2.4.13.5

Simplify the numerator.

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Step 2.4.13.5.1

Multiply $- 1$ by $2$ .

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

Step 2.4.13.5.2

Subtract $2$ from $- 1$ .

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

Step 2.4.13.6

Move the negative in front of the fraction.

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

Step 2.4.14

Move $x^{- \frac{3}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 2.4.15

Multiply $\frac{1}{2}$ by $\frac{1}{2 x^{\frac{3}{2}}}$ .

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

Step 2.4.16

Multiply $2$ by $2$ .

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

Step 2.5

Simplify.

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Step 2.5.1

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

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

Step 2.5.2

Combine terms.

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Step 2.5.2.1

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

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

Step 2.5.2.2

Add $\frac{2}{x^{3}}$ and $0$ .

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

Step 3

The second derivative of $f (x)$ with respect to $x$ is $\frac{2}{x^{3}} - \frac{1}{4 x^{\frac{3}{2}}}$ .

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