Calculus Examples

$\frac{\arcsin (2 x)}{x}$

Step 1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = \arcsin (2 x)$ and $g (x) = x$ .

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

Step 2

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

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Step 2.1

To apply the Chain Rule, set $u$ as $2 x$ .

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

Step 2.2

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

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

Step 2.3

Replace all occurrences of $u$ with $2 x$ .

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

Step 3

Differentiate.

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Step 3.1

Factor $2$ out of $2 x$ .

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

Step 3.2

Combine fractions.

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Step 3.2.1

Simplify the expression.

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Step 3.2.1.1

Apply the product rule to $2 (x)$ .

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

Step 3.2.1.2

Raise $2$ to the power of $2$ .

$\frac{x (\frac{1}{\sqrt{1 - (4 x^{2})}} \frac{d}{d x} [2 x]) - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.1.3

Multiply $4$ by $- 1$ .

$\frac{x (\frac{1}{\sqrt{1 - 4 x^{2}}} \frac{d}{d x} [2 x]) - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.2.2

Combine $\frac{1}{\sqrt{1 - 4 x^{2}}}$ and $x$ .

$\frac{\frac{x}{\sqrt{1 - 4 x^{2}}} \frac{d}{d x} [2 x] - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.3

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

$\frac{\frac{x}{\sqrt{1 - 4 x^{2}}} (2 \frac{d}{d x} [x]) - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.4

Combine $2$ and $\frac{x}{\sqrt{1 - 4 x^{2}}}$ .

$\frac{\frac{2 x}{\sqrt{1 - 4 x^{2}}} \frac{d}{d x} [x] - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.5

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

$\frac{\frac{2 x}{\sqrt{1 - 4 x^{2}}} \cdot 1 - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 3.6

Multiply $\frac{2 x}{\sqrt{1 - 4 x^{2}}}$ by $1$ .

$\frac{\frac{2 x}{\sqrt{1 - 4 x^{2}}} - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 4

Multiply $\frac{\frac{2 x}{\sqrt{1 - 4 x^{2}}} - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$ by $\frac{\sqrt{1 - 4 x^{2}}}{\sqrt{1 - 4 x^{2}}}$ .

$\frac{\sqrt{1 - 4 x^{2}}}{\sqrt{1 - 4 x^{2}}} \cdot \frac{\frac{2 x}{\sqrt{1 - 4 x^{2}}} - \arcsin (2 x) \frac{d}{d x} [x]}{x^{2}}$

Step 5

Simplify terms.

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Step 5.1

Combine.

$\frac{\sqrt{1 - 4 x^{2}} (\frac{2 x}{\sqrt{1 - 4 x^{2}}} - \arcsin (2 x) \frac{d}{d x} [x])}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 5.2

Apply the distributive property.

$\frac{\sqrt{1 - 4 x^{2}} \frac{2 x}{\sqrt{1 - 4 x^{2}}} + \sqrt{1 - 4 x^{2}} (- \arcsin (2 x) \frac{d}{d x} [x])}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 5.3

Cancel the common factor of $\sqrt{1 - 4 x^{2}}$ .

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Step 5.3.1

Cancel the common factor.

$\frac{\sqrt{1 - 4 x^{2}} \frac{2 x}{\sqrt{1 - 4 x^{2}}} + \sqrt{1 - 4 x^{2}} (- \arcsin (2 x) \frac{d}{d x} [x])}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 5.3.2

Rewrite the expression.

$\frac{2 x + \sqrt{1 - 4 x^{2}} (- \arcsin (2 x) \frac{d}{d x} [x])}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 6

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

$\frac{2 x + \sqrt{1 - 4 x^{2}} (- \arcsin (2 x) \cdot 1)}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 7

Multiply $- 1$ by $1$ .

$\frac{2 x + \sqrt{1 - 4 x^{2}} (- \arcsin (2 x))}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 8

Simplify.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite using the commutative property of multiplication.

$\frac{2 x - \sqrt{1 - 4 x^{2}} \arcsin (2 x)}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 8.1.2

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

$\frac{2 x - \sqrt{1^{2} - 4 x^{2}} \arcsin (2 x)}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 8.1.3

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$\frac{2 x - \sqrt{1^{2} - {(2 x)}^{2}} \arcsin (2 x)}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 8.1.4

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 $b = 2 x$ .

$\frac{2 x - \sqrt{(1 + 2 x) (1 - (2 x))} \arcsin (2 x)}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 8.1.5

Multiply $2$ by $- 1$ .

$\frac{2 x - \sqrt{(1 + 2 x) (1 - 2 x)} \arcsin (2 x)}{\sqrt{1 - 4 x^{2}} x^{2}}$

Step 8.2

Reorder terms.

$\frac{- \sqrt{(1 + 2 x) (1 - 2 x)} \arcsin (2 x) + 2 x}{x^{2} \sqrt{1 - 4 x^{2}}}$