Calculus Examples

$y = \arccos (\sqrt{1 - x})$

Step 1

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

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

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 $g (x) = 1 - x$ .

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Step 5.1

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

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

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

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

Step 5.3

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

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

Step 6

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

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

Step 7

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

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 1$ by $2$ .

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

Step 9.2

Subtract $2$ from $1$ .

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

Step 10

Combine fractions.

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Step 10.1

Move the negative in front of the fraction.

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 11

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

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

Multiply.

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Step 15.1

Multiply $- 1$ by $- 1$ .

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

Step 15.2

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

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

Step 16

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

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

Step 17

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

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

Step 18

Simplify.

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Step 18.1

Apply the distributive property.

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

Step 18.2

Combine terms.

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Step 18.2.1

Multiply $- 1$ by $1$ .

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

Step 18.2.2

Multiply $- 1$ by $- 1$ .

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

Step 18.2.3

Multiply $x$ by $1$ .

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

Step 18.2.4

Subtract $1$ from $1$ .

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

Step 18.2.5

Add $0$ and $x$ .

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

Step 18.3

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

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

Step 18.4

Combine and simplify the denominator.

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Step 18.4.1

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

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

Step 18.4.2

Move $\sqrt{x}$ .

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

Step 18.4.3

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 18.4.4

Raise $\sqrt{x}$ to the power of $1$ .

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

Step 18.4.5

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

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

Step 18.4.6

Add $1$ and $1$ .

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

Step 18.4.7

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 18.4.7.1

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

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

Step 18.4.7.2

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

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

Step 18.4.7.3

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

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

Step 18.4.7.4

Cancel the common factor of $2$ .

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Step 18.4.7.4.1

Cancel the common factor.

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

Step 18.4.7.4.2

Rewrite the expression.

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

Step 18.4.7.5

Simplify.

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

Step 18.5

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

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