Calculus Examples

$g (x) = \sqrt{1 - x^{2}} \arccos (x)$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $1$ .

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

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 11

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

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

Step 12

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

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

Step 13

Add $0$ and $\frac{d}{d x} [- x^{2}]$ .

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

Step 14

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

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

Step 15

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

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

Step 16

Simplify terms.

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Step 16.1

Multiply $2$ by $- 1$ .

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

Step 16.2

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

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

Step 16.3

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

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

Step 16.4

Factor $2$ out of $- 2 \arccos (x) x$ .

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

Step 17

Cancel the common factors.

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Step 17.1

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

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

Step 17.2

Cancel the common factor.

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

Step 17.3

Rewrite the expression.

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

Step 18

Move the negative in front of the fraction.

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

Step 19

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

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

Step 20

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

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

Step 21

Write each expression with a common denominator of $\sqrt{1 - x^{2}} {(1 - x^{2})}^{\frac{1}{2}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 21.1

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

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

Step 21.2

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

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

Step 21.3

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

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

Step 21.4

Combine the numerators over the common denominator.

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

Step 21.5

Add $1$ and $1$ .

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

Step 21.6

Cancel the common factor of $2$ .

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Step 21.6.1

Cancel the common factor.

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

Step 21.6.2

Rewrite the expression.

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

Step 21.7

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

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

Step 21.8

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

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

Step 21.9

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

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

Step 21.10

Combine the numerators over the common denominator.

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

Step 21.11

Add $1$ and $1$ .

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

Step 21.12

Cancel the common factor of $2$ .

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Step 21.12.1

Cancel the common factor.

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

Step 21.12.2

Rewrite the expression.

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

Step 22

Combine the numerators over the common denominator.

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

Step 23

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

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Step 23.1

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

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

Step 23.2

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

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

Step 23.3

Combine the numerators over the common denominator.

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

Step 23.4

Add $1$ and $1$ .

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

Step 23.5

Divide $2$ by $2$ .

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

Step 24

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

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

Step 25

Simplify.

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

Step 26

Simplify.

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Step 26.1

Apply the distributive property.

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

Step 26.2

Simplify the numerator.

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Step 26.2.1

Simplify each term.

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Step 26.2.1.1

Multiply $- 1$ by $1$ .

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

Step 26.2.1.2

Multiply $- - x^{2}$ .

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Step 26.2.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 26.2.1.2.2

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

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

Step 26.2.1.3

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

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

Step 26.2.1.4

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 $b = x$ .

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

Step 26.2.2

Reorder factors in $- 1 + x^{2} - \arccos (x) x \sqrt{(1 + x) (1 - x)}$ .

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

Step 26.3

Reorder terms.

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