Calculus Examples

$2 \arccos (\sqrt{x})$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

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

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

Step 1.2

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

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

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

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Cancel the common factor of $2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Simplify.

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

Step 5

Combine fractions.

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Step 5.1

Multiply $- 1$ by $2$ .

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

Step 5.2

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

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

Step 5.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Combine the numerators over the common denominator.

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

Step 10

Simplify the numerator.

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Step 10.1

Multiply $- 1$ by $2$ .

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

Step 10.2

Subtract $2$ from $1$ .

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

Step 11

Move the negative in front of the fraction.

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

Step 12

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

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

Step 13

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

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

Step 14

Simplify the expression.

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Step 14.1

Move $2$ to the left of $x^{- \frac{1}{2}}$ .

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

Step 14.2

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

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

Step 15

Cancel the common factor.

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

Step 16

Rewrite the expression.

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