Calculus Examples

$x + \sqrt{x + 1}$

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 $x + \sqrt{x + 1}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\sqrt{x + 1}]$ .

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

Step 1.1.2

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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Step 1.2.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 1.2.2.2

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

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

Step 1.2.2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

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

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

Step 1.2.6

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

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

Step 1.2.7

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

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

Step 1.2.8

Combine the numerators over the common denominator.

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

Step 1.2.9

Simplify the numerator.

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Step 1.2.9.1

Multiply $- 1$ by $2$ .

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

Step 1.2.9.2

Subtract $2$ from $1$ .

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

Step 1.2.10

Move the negative in front of the fraction.

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

Step 1.2.11

Add $1$ and $0$ .

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

Step 1.2.12

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

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

Step 1.2.13

Multiply $\frac{{(x + 1)}^{- \frac{1}{2}}}{2}$ by $1$ .

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

Step 1.2.14

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

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

Step 2

Find the second derivative.

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Step 2.1

Differentiate.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

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

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Step 2.2.3.1

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

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

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 $n = - 1$ .

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

Step 2.2.3.3

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

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

Step 2.2.4

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

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Step 2.2.4.1

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

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

Step 2.2.4.2

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

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

Step 2.2.4.3

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

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

Step 2.2.5

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

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

Step 2.2.6

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

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

Step 2.2.7

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

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

Step 2.2.8

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

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Step 2.2.8.1

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

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

Step 2.2.8.2

Cancel the common factor of $2$ .

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Step 2.2.8.2.1

Factor $2$ out of $- 2$ .

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

Step 2.2.8.2.2

Cancel the common factor.

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

Step 2.2.8.2.3

Rewrite the expression.

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

Step 2.2.9

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

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

Step 2.2.10

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

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

Step 2.2.11

Combine the numerators over the common denominator.

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

Step 2.2.12

Simplify the numerator.

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Step 2.2.12.1

Multiply $- 1$ by $2$ .

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

Step 2.2.12.2

Subtract $2$ from $1$ .

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

Step 2.2.13

Move the negative in front of the fraction.

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

Step 2.2.14

Add $1$ and $0$ .

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

Step 2.2.15

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

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

Step 2.2.16

Multiply $\frac{{(x + 1)}^{- \frac{1}{2}}}{2}$ by $1$ .

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

Step 2.2.17

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

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

Step 2.2.18

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

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

Step 2.2.19

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

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

Step 2.2.20

Multiply ${(x + 1)}^{\frac{1}{2}}$ by $x + 1$ by adding the exponents.

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Step 2.2.20.1

Move $x + 1$ .

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

Step 2.2.20.2

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

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Step 2.2.20.2.1

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

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

Step 2.2.20.2.2

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

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

Step 2.2.20.3

Write $1$ as a fraction with a common denominator.

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

Step 2.2.20.4

Combine the numerators over the common denominator.

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

Step 2.2.20.5

Add $2$ and $1$ .

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

Step 2.2.21

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

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

Step 2.2.22

Multiply $2$ by $2$ .

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

Step 2.3

Subtract $\frac{1}{4 {(x + 1)}^{\frac{3}{2}}}$ from $0$ .

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

Step 3

Find the third derivative.

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Step 3.1

Differentiate using the Constant Multiple Rule.

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Step 3.1.1

Since $- \frac{1}{4}$ is constant with respect to $x$ , the derivative of $- \frac{1}{4 {(x + 1)}^{\frac{3}{2}}}$ with respect to $x$ is $- \frac{1}{4} \frac{d}{d x} [\frac{1}{{(x + 1)}^{\frac{3}{2}}}]$ .

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

Step 3.1.2

Apply basic rules of exponents.

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Step 3.1.2.1

Rewrite $\frac{1}{{(x + 1)}^{\frac{3}{2}}}$ as ${({(x + 1)}^{\frac{3}{2}})}^{- 1}$ .

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

Step 3.1.2.2

Multiply the exponents in ${({(x + 1)}^{\frac{3}{2}})}^{- 1}$ .

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Step 3.1.2.2.1

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

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

Step 3.1.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

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Step 3.1.2.2.2.1

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

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

Step 3.1.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 3.1.2.2.3

Move the negative in front of the fraction.

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

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- \frac{3}{2}}$ and $g (x) = x + 1$ .

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Step 3.2.1

To apply the Chain Rule, set $u$ as $x + 1$ .

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

Step 3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - \frac{3}{2}$ .

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

Step 3.2.3

Replace all occurrences of $u$ with $x + 1$ .

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $- 1$ by $2$ .

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

Step 3.6.2

Subtract $2$ from $- 3$ .

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

Step 3.7

Combine fractions.

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Step 3.7.1

Move the negative in front of the fraction.

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

Step 3.7.2

Combine ${(x + 1)}^{- \frac{5}{2}}$ and $\frac{3}{2}$ .

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

Step 3.7.3

Simplify the expression.

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Step 3.7.3.1

Move $3$ to the left of ${(x + 1)}^{- \frac{5}{2}}$ .

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

Step 3.7.3.2

Move ${(x + 1)}^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.7.3.3

Multiply $- 1$ by $- 1$ .

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

Step 3.7.3.4

Multiply $\frac{1}{4}$ by $1$ .

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

Step 3.7.4

Multiply $\frac{3}{2 {(x + 1)}^{\frac{5}{2}}}$ by $\frac{1}{4}$ .

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

Step 3.7.5

Multiply $4$ by $2$ .

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

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

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

Step 3.11

Simplify the expression.

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Step 3.11.1

Add $1$ and $0$ .

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

Step 3.11.2

Multiply $\frac{3}{8 {(x + 1)}^{\frac{5}{2}}}$ by $1$ .

$f''' (x) = \frac{3}{8 {(x + 1)}^{\frac{5}{2}}}$