Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

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 $u_{2}$ as $1 - x^{2}$ .

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $1$ .

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

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Multiply.

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Step 15.1

Multiply $- 1$ by $- 1$ .

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

Step 15.2

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

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

Step 16

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

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

Step 17

Simplify terms.

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Step 17.1

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

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

Step 17.2

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

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

Step 17.3

Cancel the common factor.

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

Step 17.4

Rewrite the expression.

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

Step 18

Simplify.

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Step 18.1

Apply the distributive property.

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

Step 18.2

Combine terms.

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Step 18.2.1

Multiply $- 1$ by $1$ .

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

Step 18.2.2

Multiply $- 1$ by $- 1$ .

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

Step 18.2.3

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

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

Step 18.2.4

Subtract $1$ from $1$ .

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

Step 18.2.5

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

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

Step 18.2.6

Pull terms out from under the radical, assuming positive real numbers.

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

Step 18.2.7

Cancel the common factor.

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

Step 18.2.8

Rewrite the expression.

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

Step 18.3

Reorder terms.

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