Calculus Examples

$x arctanh (x) + \ln (\sqrt{1 - x^{2}})$

Step 1

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

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

Step 2

Evaluate $\frac{d}{d x} [x arctanh (x)]$ .

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Step 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 $f (x) = x$ and $g (x) = arctanh (x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Multiply $arctanh (x)$ by $1$ .

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

Step 2.6

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

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 3

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

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Step 3.1

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

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

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 $f (x) = \ln (x)$ and $g (x) = {(1 - x^{2})}^{\frac{1}{2}}$ .

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Step 3.2.1

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

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

Step 3.2.2

The derivative of $\ln (u_{1})$ with respect to $u_{1}$ is $\frac{1}{u_{1}}$ .

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

Step 3.2.3

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

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

Step 3.3

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

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Step 3.3.1

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

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

Step 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 $n = \frac{1}{2}$ .

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

Step 3.3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Combine the numerators over the common denominator.

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

Step 3.11

Simplify the numerator.

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Step 3.11.1

Multiply $- 1$ by $2$ .

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

Step 3.11.2

Subtract $2$ from $1$ .

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

Step 3.12

Move the negative in front of the fraction.

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

Step 3.13

Multiply $2$ by $- 1$ .

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

Step 3.14

Subtract $2 x$ from $0$ .

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

Step 3.15

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

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

Step 3.16

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

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

Step 3.17

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

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

Step 3.18

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

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

Step 3.19

Factor $2$ out of $- 2 x$ .

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

Step 3.20

Cancel the common factors.

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Step 3.20.1

Factor $2$ out of $2 {(1 - x^{2})}^{\frac{1}{2}}$ .

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

Step 3.20.2

Cancel the common factor.

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

Step 3.20.3

Rewrite the expression.

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

Step 3.21

Move the negative in front of the fraction.

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

Step 3.22

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

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

Step 3.23

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

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Step 3.23.1

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

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

Step 3.23.2

Combine the numerators over the common denominator.

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

Step 3.23.3

Add $1$ and $1$ .

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

Step 3.23.4

Divide $2$ by $2$ .

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

Step 3.24

Simplify ${(1 - x^{2})}^{1}$ .

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine terms.

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Step 4.2.1

Multiply $arctanh (x)$ by $1$ .

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

Step 4.2.2

Combine the numerators over the common denominator.

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

Step 4.2.3

Subtract $x$ from $x$ .

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

Step 4.2.4

Add $arctanh (x) + arctanh (x) (- x^{2})$ and $0$ .

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

Step 4.3

Reorder terms.

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

Step 4.4

Simplify the numerator.

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Step 4.4.1

Factor $arctanh (x)$ out of $- x^{2} arctanh (x) + arctanh (x)$ .

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Step 4.4.1.1

Factor $arctanh (x)$ out of $- x^{2} arctanh (x)$ .

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

Step 4.4.1.2

Multiply by $1$ .

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

Step 4.4.1.3

Factor $arctanh (x)$ out of $arctanh (x) (- x^{2}) + arctanh (x) \cdot 1$ .

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

Step 4.4.2

Rewrite $1$ as $1^{2}$ .

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

Step 4.4.3

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 4.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

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

Step 4.5

Simplify the denominator.

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Step 4.5.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.5.2

Reorder $- x^{2}$ and $1^{2}$ .

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

Step 4.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\frac{arctanh (x) (1 + x) (1 - x)}{(1 + x) (1 - x)}$

Step 4.6

Cancel the common factor of $1 + x$ .

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Step 4.6.1

Cancel the common factor.

$\frac{arctanh (x) (1 + x) (1 - x)}{(1 + x) (1 - x)}$

Step 4.6.2

Rewrite the expression.

$\frac{arctanh (x) (1 - x)}{1 - x}$

Step 4.7

Cancel the common factor of $1 - x$ .

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Step 4.7.1

Cancel the common factor.

$\frac{arctanh (x) (1 - x)}{1 - x}$

Step 4.7.2

Divide $arctanh (x)$ by $1$ .

$arctanh (x)$