Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

The derivative of $arcsec (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1} \sqrt{{u_{1}}^{2} - 1}}$ .

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

Simplify by subtracting numbers.

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Step 5.1

Subtract $1$ from $1$ .

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

Step 5.2

Add $x$ and $0$ .

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

Step 6

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 6.1

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

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

Step 6.2

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

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

Step 6.3

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify the expression.

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Step 16.1

Combine the numerators over the common denominator.

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

Step 16.2

Add $1$ and $1$ .

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

Step 17

Cancel the common factor of $2$ .

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Step 17.1

Cancel the common factor.

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

Step 17.2

Rewrite the expression.

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

Step 18

Simplify.

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

Step 19

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

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

Step 20

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

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

Step 21

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

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

Step 22

Simplify the expression.

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Step 22.1

Add $1$ and $0$ .

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

Step 22.2

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

$\frac{1}{2 (x + 1) \sqrt{x}}$

Step 23

Simplify.

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Step 23.1

Apply the distributive property.

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

Step 23.2

Apply the distributive property.

$\frac{1}{2 x \sqrt{x} + 2 \cdot 1 \sqrt{x}}$

Step 23.3

Combine terms.

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Step 23.3.1

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

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

Step 23.3.2

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

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Step 23.3.2.1

Move $x^{\frac{1}{2}}$ .

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

Step 23.3.2.2

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

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Step 23.3.2.2.1

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

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

Step 23.3.2.2.2

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

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

Step 23.3.2.3

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

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

Step 23.3.2.4

Combine the numerators over the common denominator.

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

Step 23.3.2.5

Add $1$ and $2$ .

$\frac{1}{2 x^{\frac{3}{2}} + 2 \cdot 1 \sqrt{x}}$

Step 23.3.3

Multiply $2$ by $1$ .

$\frac{1}{2 x^{\frac{3}{2}} + 2 \sqrt{x}}$